Up to linear equivalence, there are total 119 semisimple subalgebras (including the full subalgebra). The subalgebras are ordered by rank, Dynkin indices of simple constituents and dimensions of simple constituents. The upper index indicates the Dynkin index, the lower index indicates the rank of the subalgebra.
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (1, 2, 2, 3, 2, 1): 2 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-36}\) Positive simple generators: \(\displaystyle g_{36}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 20V_{\omega_{1}}\oplus 35V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\psi_{1}+2\psi_{5}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{3}+2\psi_{5}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+2\psi_{3}-2\psi_{4}+2\psi_{5}} \oplus V_{\omega_{1}+2\psi_{2}-2\psi_{4}+2\psi_{5}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{2}-2\psi_{3}+2\psi_{4}} \oplus V_{\omega_{1}+2\psi_{3}}\oplus V_{-2\psi_{4}+4\psi_{5}}\oplus V_{-2\psi_{3}+2\psi_{4}+2\psi_{5}}\oplus V_{-2\psi_{2}+2\psi_{3}+2\psi_{5}} \oplus V_{-2\psi_{1}+2\psi_{2}+2\psi_{5}}\oplus V_{2\omega_{1}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{2\psi_{1}+2\psi_{2}-2\psi_{3}} \oplus V_{2\psi_{1}+2\psi_{3}-2\psi_{4}}\oplus V_{2\psi_{1}+2\psi_{4}-2\psi_{5}}\oplus V_{\omega_{1}-2\psi_{2}+2\psi_{5}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-2\psi_{3}+2\psi_{5}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{3}-2\psi_{4}+2\psi_{5}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{4}} \oplus V_{\omega_{1}+2\psi_{1}-2\psi_{3}+2\psi_{4}-2\psi_{5}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+2\psi_{3}-2\psi_{5}} \oplus V_{\omega_{1}+2\psi_{2}-2\psi_{5}}\oplus V_{2\psi_{1}-2\psi_{2}-2\psi_{4}+2\psi_{5}}\oplus V_{2\psi_{2}-2\psi_{3}-2\psi_{4}+2\psi_{5}} \oplus V_{2\psi_{3}-4\psi_{4}+2\psi_{5}}\oplus V_{2\psi_{1}-2\psi_{2}-2\psi_{3}+2\psi_{4}}\oplus V_{2\psi_{2}-4\psi_{3}+2\psi_{4}} \oplus V_{2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus 5V_{0}\oplus V_{-2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{2}+4\psi_{3}-2\psi_{4}} \oplus V_{-2\psi_{1}+2\psi_{2}+2\psi_{3}-2\psi_{4}}\oplus V_{-2\psi_{3}+4\psi_{4}-2\psi_{5}}\oplus V_{-2\psi_{2}+2\psi_{3}+2\psi_{4}-2\psi_{5}} \oplus V_{-2\psi_{1}+2\psi_{2}+2\psi_{4}-2\psi_{5}}\oplus V_{\omega_{1}-2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{2}+2\psi_{3}-2\psi_{4}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-2\psi_{4}}\oplus V_{\omega_{1}-2\psi_{2}+2\psi_{4}-2\psi_{5}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-2\psi_{3}+2\psi_{4}-2\psi_{5}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{3}-2\psi_{5}}\oplus V_{-2\psi_{1}-2\psi_{4}+2\psi_{5}}\oplus V_{-2\psi_{1}-2\psi_{3}+2\psi_{4}} \oplus V_{-2\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-2\psi_{2}-2\psi_{5}}\oplus V_{2\psi_{2}-2\psi_{3}-2\psi_{5}} \oplus V_{2\psi_{3}-2\psi_{4}-2\psi_{5}}\oplus V_{2\psi_{4}-4\psi_{5}}\oplus V_{-2\psi_{1}-2\psi_{5}}\) Made total 282 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1\) ↪ \(E^{1}_6\) 2 out of 119
Subalgebra type: \(\displaystyle A^{2}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle B^{1}_3\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_1\) Basis of Cartan of centralizer: 4 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_1+A^{1}_1\) , \(\displaystyle 2A^{2}_1\) , \(\displaystyle A^{3}_1+A^{2}_1\) , \(\displaystyle A^{4}_1+A^{2}_1\) , \(\displaystyle A^{10}_1+A^{2}_1\) , \(\displaystyle A^{28}_1+A^{2}_1\) , \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle 2A^{2}_1+A^{1}_1\) , \(\displaystyle 3A^{2}_1\) , \(\displaystyle A^{3}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle A^{1}_2+A^{2}_1\) , \(\displaystyle B^{1}_2+A^{2}_1\) , \(\displaystyle G^{1}_2+A^{2}_1\) , \(\displaystyle 2A^{2}_1+2A^{1}_1\) , \(\displaystyle A^{1}_3+A^{2}_1\) , \(\displaystyle B^{1}_3+A^{2}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 2, 3, 4, 3, 2): 4 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 8V_{2\omega_{1}}\oplus 16V_{\omega_{1}}\oplus 22V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{2}+2\psi_{3}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}+2\psi_{3}+2\psi_{4}}\oplus V_{2\psi_{1}+2\psi_{2}+4\psi_{3}-2\psi_{4}} \oplus V_{2\omega_{1}+2\psi_{2}+4\psi_{3}-2\psi_{4}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}+2\psi_{3}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{3}+2\psi_{4}} \oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{-2\psi_{1}+4\psi_{2}+4\psi_{3}-2\psi_{4}}\oplus V_{2\psi_{2}+4\psi_{3}-2\psi_{4}} \oplus V_{2\psi_{1}+4\psi_{3}-2\psi_{4}}\oplus V_{\omega_{1}-2\psi_{3}+4\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}+4\psi_{4}} \oplus V_{2\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}}\oplus 2V_{2\omega_{1}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}} \oplus V_{-2\psi_{1}+2\psi_{2}+4\psi_{3}-2\psi_{4}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{3}+4\psi_{4}}\oplus V_{\omega_{1}-2\psi_{2}-2\psi_{3}+4\psi_{4}} \oplus V_{\omega_{1}+2\psi_{2}+2\psi_{3}-4\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}+2\psi_{3}-4\psi_{4}}\oplus V_{-2\psi_{1}+2\psi_{2}} \oplus 4V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}+2\psi_{3}-4\psi_{4}} \oplus V_{\omega_{1}+2\psi_{3}-4\psi_{4}}\oplus V_{2\psi_{1}-2\psi_{2}-4\psi_{3}+2\psi_{4}}\oplus V_{2\omega_{1}-2\psi_{2}-4\psi_{3}+2\psi_{4}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{3}-2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}-2\psi_{4}} \oplus V_{-2\psi_{1}-4\psi_{3}+2\psi_{4}}\oplus V_{-2\psi_{2}-4\psi_{3}+2\psi_{4}}\oplus V_{2\psi_{1}-4\psi_{2}-4\psi_{3}+2\psi_{4}} \oplus V_{\omega_{1}-2\psi_{1}-2\psi_{3}-2\psi_{4}}\oplus V_{\omega_{1}-2\psi_{2}-2\psi_{3}-2\psi_{4}}\oplus V_{-2\psi_{1}-2\psi_{2}-4\psi_{3}+2\psi_{4}}\) Made total 6531159 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1\) ↪ \(E^{1}_6\) 3 out of 119
Subalgebra type: \(\displaystyle A^{3}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{2}_2+A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{3}_1\) Basis of Cartan of centralizer: 3 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+A^{1}_1\) , \(\displaystyle A^{3}_1+A^{2}_1\) , \(\displaystyle 2A^{3}_1\) , \(\displaystyle A^{8}_1+A^{3}_1\) , \(\displaystyle A^{9}_1+A^{3}_1\) , \(\displaystyle A^{3}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{3}_1\) , \(\displaystyle A^{2}_2+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 3, 4, 6, 4, 2): 6 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-23}+g_{-30}+g_{-34}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}+g_{23}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{3\omega_{1}}\oplus 9V_{2\omega_{1}}\oplus 16V_{\omega_{1}}\oplus 11V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{3\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{2\omega_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{2\psi_{2}+2\psi_{3}}\oplus V_{4\psi_{1}}\oplus V_{2\omega_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}+2\psi_{3}} \oplus 2V_{\omega_{1}+2\psi_{1}}\oplus V_{-2\psi_{2}+4\psi_{3}}\oplus 3V_{2\omega_{1}}\oplus V_{4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{\omega_{1}+2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{3\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}-2\psi_{1}+4\psi_{2}-2\psi_{3}} \oplus V_{\omega_{1}+2\psi_{1}+2\psi_{2}-4\psi_{3}}\oplus V_{2\omega_{1}-4\psi_{2}+2\psi_{3}}\oplus 3V_{0}\oplus V_{2\omega_{1}+2\psi_{2}-4\psi_{3}} \oplus 2V_{\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{1}-2\psi_{2}-2\psi_{3}} \oplus V_{2\psi_{2}-4\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{-4\psi_{1}}\oplus V_{-2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}-2\psi_{3}}\) Made total 5853604 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1\) ↪ \(E^{1}_6\) 4 out of 119
Subalgebra type: \(\displaystyle A^{4}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle 2A^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_2\) Basis of Cartan of centralizer: 4 vectors:
(0, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{4}_1+A^{1}_1\) , \(\displaystyle A^{4}_1+A^{2}_1\) , \(\displaystyle 2A^{4}_1\) , \(\displaystyle A^{5}_1+A^{4}_1\) , \(\displaystyle A^{8}_1+A^{4}_1\) , \(\displaystyle A^{4}_1+2A^{1}_1\) , \(\displaystyle 2A^{4}_1+A^{1}_1\) , \(\displaystyle 3A^{4}_1\) , \(\displaystyle A^{1}_2+A^{4}_1\) , \(\displaystyle A^{2}_2+A^{4}_1\) , \(\displaystyle A^{2}_2+A^{4}_1\) , \(\displaystyle A^{1}_2+A^{4}_1+A^{1}_1\) , \(\displaystyle A^{1}_2+2A^{4}_1\) , \(\displaystyle 2A^{1}_2+A^{4}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 4, 6, 4, 2): 8 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-23}+g_{-27}\) Positive simple generators: \(\displaystyle 2g_{27}+2g_{23}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}}\oplus 19V_{2\omega_{1}}\oplus 16V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\psi_{2}+2\psi_{3}+4\psi_{4}}\oplus V_{2\omega_{1}+2\psi_{3}+4\psi_{4}}\oplus V_{2\omega_{1}-2\psi_{1}+4\psi_{2}+4\psi_{4}} \oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}+2\psi_{3}+2\psi_{4}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{4}}\oplus V_{-2\psi_{1}+2\psi_{2}+4\psi_{4}} \oplus V_{-2\psi_{2}+4\psi_{3}+2\psi_{4}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{3}+2\psi_{4}}\oplus V_{4\psi_{2}-2\psi_{3}+2\psi_{4}} \oplus V_{2\omega_{1}+2\psi_{2}-2\psi_{3}+2\psi_{4}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{-4\psi_{1}+4\psi_{2}+2\psi_{4}}\oplus V_{2\psi_{1}-2\psi_{2}+2\psi_{4}}\oplus V_{2\omega_{1}+2\psi_{1}-4\psi_{2}+2\psi_{3}} \oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}-2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus 4V_{0}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}-2\psi_{3}} \oplus V_{2\omega_{1}-2\psi_{2}+2\psi_{3}-2\psi_{4}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{3}-2\psi_{4}}\oplus V_{-2\psi_{1}+2\psi_{2}-2\psi_{4}} \oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{4}}\oplus V_{4\psi_{1}-4\psi_{2}-2\psi_{4}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}-2\psi_{4}} \oplus V_{-4\psi_{2}+2\psi_{3}-2\psi_{4}}\oplus V_{2\psi_{2}-4\psi_{3}-2\psi_{4}}\oplus V_{2\psi_{1}-2\psi_{2}-4\psi_{4}}\oplus V_{2\omega_{1}+2\psi_{1}-4\psi_{2}-4\psi_{4}} \oplus V_{2\omega_{1}-2\psi_{3}-4\psi_{4}}\oplus V_{-2\psi_{2}-2\psi_{3}-4\psi_{4}}\) Made total 45203957 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_1\) ↪ \(E^{1}_6\) 5 out of 119
Subalgebra type: \(\displaystyle A^{5}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_2\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_2\) Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0), (1, 0, 0, 2, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{5}_1+A^{1}_1\) , \(\displaystyle A^{5}_1+A^{4}_1\) , \(\displaystyle A^{1}_2+A^{5}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_1\): (3, 4, 5, 7, 5, 3): 10 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-22}+g_{-24}+g_{-28}\) Positive simple generators: \(\displaystyle 2g_{28}+g_{24}+2g_{22}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}}\oplus 6V_{3\omega_{1}}\oplus 8V_{2\omega_{1}}\oplus 8V_{\omega_{1}}\oplus 9V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}-2\psi_{1}+2\psi_{2}+8\psi_{3}}\oplus V_{3\omega_{1}+2\psi_{2}+4\psi_{3}}\oplus V_{-2\psi_{1}+4\psi_{2}+6\psi_{3}} \oplus V_{\omega_{1}+6\psi_{3}}\oplus V_{\omega_{1}+2\psi_{2}+4\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{3}}\oplus V_{3\omega_{1}-2\psi_{1}+4\psi_{3}} \oplus V_{3\omega_{1}-2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}+6\psi_{3}}\oplus V_{2\psi_{1}+2\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{\omega_{1}-2\psi_{1}+4\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{1}-2\psi_{2}+2\psi_{3}} \oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{3\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{3\omega_{1}+2\psi_{1}-4\psi_{3}} \oplus 3V_{0}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-4\psi_{3}}\oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{3}} \oplus V_{3\omega_{1}-2\psi_{2}-4\psi_{3}}\oplus V_{-2\psi_{1}-2\psi_{2}}\oplus V_{4\psi_{1}-2\psi_{2}-6\psi_{3}}\oplus V_{\omega_{1}-2\psi_{2}-4\psi_{3}} \oplus V_{\omega_{1}-6\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}-8\psi_{3}}\oplus V_{2\psi_{1}-4\psi_{2}-6\psi_{3}}\) Made total 9538973 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{6}_1\) ↪ \(E^{1}_6\) 6 out of 119
Subalgebra type: \(\displaystyle A^{6}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{6}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{6}_1\) Basis of Cartan of centralizer: 2 vectors:
(1, -1, 1, -1, -1, 0), (1, -2, 0, -2, 0, 1) Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{6}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{6}_1\): (3, 4, 6, 8, 6, 3): 12 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-15}+g_{-22}+g_{-25}+g_{-30}\) Positive simple generators: \(\displaystyle 2g_{30}+g_{25}+g_{22}+2g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}12\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus 9V_{2\omega_{1}}\oplus 8V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi_{1}+8\psi_{2}}\oplus V_{\omega_{1}+6\psi_{1}+6\psi_{2}}\oplus V_{4\omega_{1}+2\psi_{1}+4\psi_{2}} \oplus V_{3\omega_{1}+4\psi_{1}+2\psi_{2}}\oplus 2V_{2\omega_{1}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+6\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}+2\psi_{2}} \oplus V_{2\psi_{1}+4\psi_{2}}\oplus V_{4\omega_{1}}\oplus V_{3\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+2\psi_{1}-2\psi_{2}} \oplus 3V_{2\omega_{1}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{4\omega_{1}-2\psi_{1}-4\psi_{2}} \oplus V_{3\omega_{1}-4\psi_{1}-2\psi_{2}}\oplus 2V_{2\omega_{1}-2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}-4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}-6\psi_{2}}\oplus V_{-2\psi_{1}-4\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}-8\psi_{2}}\oplus V_{\omega_{1}-6\psi_{1}-6\psi_{2}}\) Made total 220476270 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1\) ↪ \(E^{1}_6\) 7 out of 119
Subalgebra type: \(\displaystyle A^{8}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle G^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 1, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1\) , \(\displaystyle A^{8}_1+A^{4}_1\) , \(\displaystyle A^{28}_1+A^{8}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{1}_2+A^{8}_1\) , \(\displaystyle G^{1}_2+A^{8}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 4, 6, 8, 6, 4): 16 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-7}+g_{-11}+g_{-26}+g_{-28}\) Positive simple generators: \(\displaystyle 2g_{28}+2g_{26}+2g_{11}+2g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 8V_{4\omega_{1}}\oplus 8V_{2\omega_{1}}\oplus 14V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\psi_{2}}\oplus V_{4\omega_{1}+2\psi_{1}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{2}}\oplus V_{4\omega_{1}-2\psi_{1}+2\psi_{2}} \oplus V_{2\omega_{1}+2\psi_{1}}\oplus 2V_{4\omega_{1}}\oplus V_{4\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}} \oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}}\oplus 2V_{2\omega_{1}}\oplus V_{4\omega_{1}-2\psi_{1}}\oplus V_{4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}} \oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}} \oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 22819045 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1\) ↪ \(E^{1}_6\) 8 out of 119
Subalgebra type: \(\displaystyle A^{9}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{3}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_2+A^{1}_1\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, -1, 0, -1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{9}_1+A^{3}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (4, 5, 7, 10, 7, 4): 18 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-12}+g_{-16}+g_{-22}+g_{-23}+g_{-25}\) Positive simple generators: \(\displaystyle 2g_{25}+g_{23}+2g_{22}+2g_{16}+2g_{12}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{5\omega_{1}}\oplus 4V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus 5V_{2\omega_{1}}\oplus 6V_{\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi}\oplus V_{\omega_{1}+6\psi}\oplus V_{5\omega_{1}+2\psi}\oplus V_{2\omega_{1}+4\psi}\oplus 2V_{3\omega_{1}+2\psi} \oplus V_{4\psi}\oplus 2V_{4\omega_{1}}\oplus 2V_{\omega_{1}+2\psi}\oplus V_{5\omega_{1}-2\psi}\oplus 3V_{2\omega_{1}}\oplus 2V_{3\omega_{1}-2\psi} \oplus V_{0}\oplus V_{4\omega_{1}-4\psi}\oplus 2V_{\omega_{1}-2\psi}\oplus V_{2\omega_{1}-4\psi}\oplus V_{-4\psi}\oplus V_{\omega_{1}-6\psi}\) Made total 3644384 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1\) ↪ \(E^{1}_6\) 9 out of 119
Subalgebra type: \(\displaystyle A^{10}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle B^{1}_2\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\) Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{2}_1\) , \(\displaystyle 2A^{10}_1\) , \(\displaystyle A^{10}_1+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{10}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (4, 6, 7, 10, 7, 4): 20 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-8}+g_{-19}+g_{-24}\) Positive simple generators: \(\displaystyle 4g_{24}+3g_{19}+3g_{8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 5V_{4\omega_{1}}\oplus 8V_{3\omega_{1}}\oplus V_{2\omega_{1}}\oplus 11V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{4\omega_{1}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{3\omega_{1}+2\psi_{2}+2\psi_{3}} \oplus V_{4\omega_{1}+2\psi_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{3\omega_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{4\omega_{1}}\oplus V_{2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{3\omega_{1}-2\psi_{1}-2\psi_{2}+4\psi_{3}}\oplus V_{3\omega_{1}+2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{2\psi_{1}}\oplus V_{2\omega_{1}}\oplus V_{4\omega_{1}-2\psi_{1}}\oplus V_{4\psi_{2}-2\psi_{3}}\oplus V_{3\omega_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{4\omega_{1}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus 3V_{0}\oplus V_{3\omega_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{2}+2\psi_{3}} \oplus V_{-2\psi_{1}}\oplus V_{3\omega_{1}-2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{-4\psi_{1}-4\psi_{2}+2\psi_{3}}\) Made total 4613803 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1\) ↪ \(E^{1}_6\) 10 out of 119
Subalgebra type: \(\displaystyle A^{11}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2) Contained up to conjugation as a direct summand of: \(\displaystyle A^{11}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (4, 6, 8, 11, 8, 4): 22 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-13}+g_{-14}+g_{-15}+g_{-24}\) Positive simple generators: \(\displaystyle 4g_{24}+g_{15}+3g_{14}+3g_{13}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 6V_{3\omega_{1}}\oplus 4V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+2\psi_{1}+6\psi_{2}}\oplus V_{4\omega_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+6\psi_{2}}\oplus V_{3\omega_{1}-2\psi_{1}+6\psi_{2}} \oplus V_{5\omega_{1}+2\psi_{1}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi_{1}}\oplus V_{4\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{\omega_{1}+2\psi_{1}} \oplus V_{5\omega_{1}-2\psi_{1}}\oplus 2V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{3\omega_{1}+2\psi_{1}-6\psi_{2}} \oplus V_{4\omega_{1}-6\psi_{2}}\oplus V_{-4\psi_{1}}\oplus V_{2\omega_{1}-6\psi_{2}}\oplus V_{3\omega_{1}-2\psi_{1}-6\psi_{2}}\) Made total 2168003 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{12}_1\) ↪ \(E^{1}_6\) 11 out of 119
Subalgebra type: \(\displaystyle A^{12}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{12}_1\): (4, 6, 8, 12, 8, 4): 24 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-4}-2g_{-15}+g_{-19}+g_{-27}\) Positive simple generators: \(\displaystyle 4g_{27}+g_{24}+2g_{19}-g_{15}+4g_{8}+4g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/6\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}24\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{6\omega_{1}}\oplus 7V_{4\omega_{1}}\oplus 9V_{2\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{4\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{2}} \oplus 2V_{6\omega_{1}}\oplus V_{4\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{4\omega_{1}} \oplus V_{2\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus 3V_{2\omega_{1}}\oplus V_{4\omega_{1}+2\psi_{1}-4\psi_{2}} \oplus V_{2\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{4\omega_{1}-2\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}-4\psi_{2}} \oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{2}}\) Made total 583010629 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{20}_1\) ↪ \(E^{1}_6\) 12 out of 119
Subalgebra type: \(\displaystyle A^{20}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (1, 0, -1, 0, 1, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{20}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{20}_1\): (6, 8, 10, 14, 10, 6): 40 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-7}+g_{-8}+g_{-16}+g_{-19}\) Positive simple generators: \(\displaystyle 4g_{19}+6g_{16}+4g_{8}+6g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/10\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}40\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{8\omega_{1}}\oplus 5V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 5V_{2\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+6\psi_{2}}\oplus V_{6\omega_{1}-2\psi_{1}+6\psi_{2}}\oplus V_{6\omega_{1}+2\psi_{1}}\oplus V_{8\omega_{1}}\oplus V_{2\omega_{1}-2\psi_{1}+6\psi_{2}} \oplus V_{6\omega_{1}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{6\omega_{1}-2\psi_{1}}\oplus V_{-4\psi_{1}+6\psi_{2}} \oplus V_{2\omega_{1}}\oplus V_{6\omega_{1}+2\psi_{1}-6\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{4\psi_{1}-6\psi_{2}} \oplus V_{2\omega_{1}+2\psi_{1}-6\psi_{2}}\oplus V_{4\omega_{1}-6\psi_{2}}\) Made total 17521203 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{21}_1\) ↪ \(E^{1}_6\) 13 out of 119
Subalgebra type: \(\displaystyle A^{21}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 3, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{21}_1\): (6, 8, 11, 15, 11, 6): 42 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-7}+g_{-13}+g_{-14}+g_{-15}+g_{-16}\) Positive simple generators: \(\displaystyle 6g_{16}+g_{15}+4g_{14}+4g_{13}+6g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/21\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}42\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{8\omega_{1}}\oplus 2V_{7\omega_{1}}\oplus V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus 2V_{2\omega_{1}} \oplus 2V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+12\psi}\oplus V_{7\omega_{1}+6\psi}\oplus V_{5\omega_{1}+6\psi}\oplus V_{3\omega_{1}+6\psi}\oplus V_{8\omega_{1}}\oplus V_{\omega_{1}+6\psi} \oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}}\oplus V_{7\omega_{1}-6\psi}\oplus V_{0}\oplus V_{5\omega_{1}-6\psi}\oplus V_{3\omega_{1}-6\psi} \oplus V_{\omega_{1}-6\psi}\oplus V_{4\omega_{1}-12\psi}\) Made total 2116053 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{28}_1\) ↪ \(E^{1}_6\) 14 out of 119
Subalgebra type: \(\displaystyle A^{28}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{2}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{28}_1+A^{2}_1\) , \(\displaystyle A^{28}_1+A^{8}_1\) , \(\displaystyle A^{2}_2+A^{28}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{28}_1\): (6, 10, 12, 18, 12, 6): 56 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-2}+g_{-4}+g_{-15}+g_{-24}\) Positive simple generators: \(\displaystyle 6g_{24}+6g_{15}+6g_{4}+10g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 8V_{6\omega_{1}}\oplus V_{2\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{6\omega_{1}+4\psi_{1}-2\psi_{2}} \oplus 2V_{6\omega_{1}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{6\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{6\omega_{1}+2\psi_{1}-4\psi_{2}} \oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{6\omega_{1}-2\psi_{1}-2\psi_{2}}\oplus 2V_{0} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 9668148 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{30}_1\) ↪ \(E^{1}_6\) 15 out of 119
Subalgebra type: \(\displaystyle A^{30}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{30}_1\): (7, 10, 13, 18, 13, 7): 60 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-2}+g_{-7}+1/2g_{-12}+g_{-15}+g_{-16}\) Positive simple generators: \(\displaystyle 7g_{16}+6g_{15}+10g_{12}+g_{11}-10g_{8}+2g_{7}+10g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/15\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}60\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus 2V_{7\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}} \oplus 2V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{7\omega_{1}+6\psi}\oplus V_{5\omega_{1}+6\psi}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{\omega_{1}+6\psi}\oplus 2V_{6\omega_{1}} \oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}}\oplus V_{7\omega_{1}-6\psi}\oplus V_{0}\oplus V_{5\omega_{1}-6\psi}\oplus V_{\omega_{1}-6\psi}\) Made total 2337651 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1\) ↪ \(E^{1}_6\) 16 out of 119
Subalgebra type: \(\displaystyle A^{35}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{35}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (8, 10, 14, 19, 14, 8): 70 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-6}+g_{-13}+g_{-14}+g_{-15}\) Positive simple generators: \(\displaystyle 9g_{15}+5g_{14}+5g_{13}+8g_{6}+8g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 2V_{9\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus V_{4\omega_{1}}\oplus 2V_{3\omega_{1}} \oplus V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{9\omega_{1}+2\psi}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{5\omega_{1}+2\psi}\oplus V_{9\omega_{1}-2\psi}\oplus V_{6\omega_{1}} \oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}}\oplus V_{5\omega_{1}-2\psi}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi} \oplus V_{0}\oplus V_{-4\psi}\) Made total 59553 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{36}_1\) ↪ \(E^{1}_6\) 17 out of 119
Subalgebra type: \(\displaystyle A^{36}_1\) (click on type for detailed printout).
Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{36}_1\): (8, 10, 14, 20, 14, 8): 72 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-6}+g_{-8}+g_{-9}+g_{-10}+g_{-19}\) Positive simple generators: \(\displaystyle 9g_{19}+5g_{10}+5g_{9}+g_{8}+8g_{6}+8g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/18\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}72\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{10\omega_{1}}\oplus 2V_{8\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\) Made total 14218918 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{60}_1\) ↪ \(E^{1}_6\) 18 out of 119
Subalgebra type: \(\displaystyle A^{60}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{60}_1\): (10, 14, 18, 26, 18, 10): 120 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-4}+g_{-6}+g_{-15}\) Positive simple generators: \(\displaystyle 18g_{15}+10g_{6}+8g_{4}+14g_{2}+10g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/30\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}120\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus 3V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus 2V_{4\omega_{1}}\oplus V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{10\omega_{1}+6\psi}\oplus V_{14\omega_{1}}\oplus V_{4\omega_{1}+6\psi}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}} \oplus V_{10\omega_{1}-6\psi}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{4\omega_{1}-6\psi}\) Made total 3102109 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{84}_1\) ↪ \(E^{1}_6\) 19 out of 119
Subalgebra type: \(\displaystyle A^{84}_1\) (click on type for detailed printout).
Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{84}_1\): (12, 16, 22, 30, 22, 12): 168 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-6}+g_{-9}+g_{-10}\) Positive simple generators: \(\displaystyle 22g_{10}+8g_{9}-8g_{8}+12g_{6}+14g_{5}+14g_{3}+16g_{2}+12g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/42\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}168\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{16\omega_{1}}\oplus V_{14\omega_{1}}\oplus 2V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\) Made total 3582365 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{156}_1\) ↪ \(E^{1}_6\) 20 out of 119
Subalgebra type: \(\displaystyle A^{156}_1\) (click on type for detailed printout).
Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{156}_1\): (16, 22, 30, 42, 30, 16): 312 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}\) Positive simple generators: \(\displaystyle 16g_{6}+30g_{5}+42g_{4}+30g_{3}+22g_{2}+16g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/78\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}312\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{22\omega_{1}}\oplus V_{16\omega_{1}}\oplus V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{2\omega_{1}}\) Made total 129084 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{1}_1\) ↪ \(E^{1}_6\) 21 out of 119
Subalgebra type: \(\displaystyle 2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) . Centralizer: \(\displaystyle A^{1}_3\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_1\) Basis of Cartan of centralizer: 4 vectors:
(0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle 3A^{1}_1\) , \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle A^{4}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+2A^{1}_1\) , \(\displaystyle 4A^{1}_1\) , \(\displaystyle 2A^{2}_1+2A^{1}_1\) , \(\displaystyle A^{1}_2+2A^{1}_1\) , \(\displaystyle B^{1}_2+2A^{1}_1\) , \(\displaystyle A^{1}_3+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (1, 2, 2, 3, 2, 1): 2, \(\displaystyle A^{1}_1\): (1, 0, 1, 1, 1, 1): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{-24}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{24}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 6V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 8V_{\omega_{2}}\oplus 8V_{\omega_{1}}\oplus 16V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi_{3}+4\psi_{4}}\oplus V_{\omega_{1}-2\psi_{2}+2\psi_{3}+4\psi_{4}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}+4\psi_{4}} \oplus V_{\omega_{1}+2\psi_{1}+2\psi_{4}}\oplus V_{-2\psi_{2}+4\psi_{3}+2\psi_{4}}\oplus V_{-2\psi_{1}+2\psi_{2}+2\psi_{3}+2\psi_{4}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{3}+2\psi_{4}}\oplus V_{2\psi_{1}+2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}+2\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}+2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}+4\psi_{4}}\oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}+2\psi_{4}} \oplus V_{\omega_{2}+2\psi_{2}-2\psi_{3}+2\psi_{4}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}-2\psi_{3}+2\psi_{4}} \oplus V_{\omega_{1}+2\psi_{3}-2\psi_{4}}\oplus V_{-2\psi_{1}-2\psi_{2}+2\psi_{3}+2\psi_{4}}\oplus V_{-4\psi_{1}+2\psi_{2}+2\psi_{4}} \oplus V_{2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus 4V_{0}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}-2\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{4\psi_{1}-2\psi_{2}-2\psi_{4}}\oplus V_{2\psi_{1}+2\psi_{2}-2\psi_{3}-2\psi_{4}} \oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{3}-2\psi_{4}}\oplus V_{\omega_{2}-2\psi_{2}+2\psi_{3}-2\psi_{4}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}-2\psi_{4}} \oplus V_{\omega_{2}+2\psi_{1}-4\psi_{4}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-4\psi_{4}} \oplus V_{\omega_{1}+2\psi_{2}-2\psi_{3}-4\psi_{4}}\oplus V_{-2\psi_{1}-2\psi_{3}}\oplus V_{2\psi_{1}-2\psi_{2}-2\psi_{3}-2\psi_{4}} \oplus V_{2\psi_{2}-4\psi_{3}-2\psi_{4}}\oplus V_{\omega_{2}-2\psi_{3}-4\psi_{4}}\) Made total 373 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1+A^{1}_1\) ↪ \(E^{1}_6\) 22 out of 119
Subalgebra type: \(\displaystyle A^{2}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle A^{2}_1+A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_1+A^{1}_1\) Basis of Cartan of centralizer: 3 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle 2A^{2}_1+A^{1}_1\) , \(\displaystyle A^{3}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle 2A^{2}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 2, 3, 4, 3, 2): 4, \(\displaystyle A^{1}_1\): (0, 1, 1, 2, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{-23}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle g_{23}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 4V_{2\omega_{1}}\oplus 6V_{\omega_{2}}\oplus 8V_{\omega_{1}} \oplus 7V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}}\oplus V_{\omega_{2}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{2}+4\psi_{3}} \oplus V_{4\psi_{1}}\oplus V_{2\omega_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}} \oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}-4\psi_{2}+2\psi_{3}} \oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}}\oplus V_{\omega_{2}-2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{2\omega_{1}-4\psi_{2}+2\psi_{3}}\oplus 3V_{0}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{2}-4\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}} \oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{2}-2\psi_{3}} \oplus V_{\omega_{2}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-4\psi_{3}}\oplus V_{-4\psi_{1}} \oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}-2\psi_{3}}\) Made total 630 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{2}_1\) ↪ \(E^{1}_6\) 23 out of 119
Subalgebra type: \(\displaystyle 2A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle 2A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\) Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{2}_1+A^{1}_1\) , \(\displaystyle 3A^{2}_1\) , \(\displaystyle 2A^{2}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 2, 3, 4, 3, 2): 4, \(\displaystyle A^{2}_1\): (0, 2, 1, 2, 1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{-8}+g_{-19}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle g_{19}+g_{8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus 5V_{2\omega_{2}}\oplus 8V_{\omega_{1}+\omega_{2}}\oplus 5V_{2\omega_{1}}\oplus 7V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{2}+2\psi_{3}}\oplus V_{4\psi_{1}+4\psi_{2}-2\psi_{3}} \oplus V_{2\omega_{2}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{2}+4\psi_{3}} \oplus V_{2\omega_{2}+2\psi_{1}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus 3V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{2}-4\psi_{3}} \oplus V_{-4\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{2}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{1}-2\psi_{1}-4\psi_{2}+2\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{1}-4\psi_{2}+2\psi_{3}}\) Made total 3277348 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+A^{1}_1\) ↪ \(E^{1}_6\) 24 out of 119
Subalgebra type: \(\displaystyle A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle A^{2}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 3, 4, 6, 4, 2): 6, \(\displaystyle A^{1}_1\): (0, 1, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-23}+g_{-30}+g_{-34}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}+g_{23}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 8V_{\omega_{1}+\omega_{2}}\oplus 9V_{2\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}+4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-4\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}} \oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 1029 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+A^{2}_1\) ↪ \(E^{1}_6\) 25 out of 119
Subalgebra type: \(\displaystyle A^{3}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0, 0, 0), (1, 0, -1, 0, 1, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+A^{2}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 3, 4, 6, 4, 2): 6, \(\displaystyle A^{2}_1\): (1, 0, 1, 0, 1, 1): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-29}+g_{-30}+g_{-31}\), \(\displaystyle g_{-7}-g_{-11}\) Positive simple generators: \(\displaystyle g_{31}+g_{30}+g_{29}\), \(\displaystyle -g_{11}+g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+2\omega_{2}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{2}} \oplus 4V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+6\psi_{2}}\oplus V_{\omega_{2}+6\psi_{2}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+6\psi_{2}}\oplus V_{\omega_{1}+2\omega_{2}+2\psi_{1}}\oplus V_{3\omega_{1}+2\psi_{1}} \oplus V_{4\psi_{1}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{\omega_{1}+2\psi_{1}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}+2\omega_{2}-2\psi_{1}} \oplus V_{3\omega_{1}-2\psi_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-6\psi_{2}} \oplus V_{-4\psi_{1}}\oplus V_{\omega_{2}-6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-6\psi_{2}}\) Made total 825441 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{3}_1\) ↪ \(E^{1}_6\) 26 out of 119
Subalgebra type: \(\displaystyle 2A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 3, 4, 6, 4, 2): 6, \(\displaystyle A^{3}_1\): (1, 1, 1, 0, 1, 1): 6 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-29}+g_{-30}+g_{-31}\), \(\displaystyle g_{-2}+g_{-7}-g_{-11}\) Positive simple generators: \(\displaystyle g_{31}+g_{30}+g_{29}\), \(\displaystyle -g_{11}+g_{7}+g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0\\
0 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0\\
0 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+3\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{1}+2\omega_{2}} \oplus 2V_{2\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}} \oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\omega_{2}+6\psi}\oplus V_{2\omega_{1}+\omega_{2}+6\psi}\oplus V_{\omega_{2}+6\psi}\oplus V_{\omega_{1}+6\psi} \oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}} \oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+2\omega_{2}-6\psi}\oplus V_{2\omega_{1}+\omega_{2}-6\psi}\oplus V_{\omega_{2}-6\psi} \oplus V_{\omega_{1}-6\psi}\) Made total 1471846 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1+A^{1}_1\) ↪ \(E^{1}_6\) 27 out of 119
Subalgebra type: \(\displaystyle A^{4}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1\) . Centralizer: \(\displaystyle A^{1}_2\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_2\) Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0), (1, 0, 0, 2, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{4}_1+2A^{1}_1\) , \(\displaystyle 2A^{4}_1+A^{1}_1\) , \(\displaystyle A^{1}_2+A^{4}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 4, 6, 4, 2): 8, \(\displaystyle A^{1}_1\): (1, 0, 1, 1, 1, 1): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-22}+g_{-28}\), \(\displaystyle g_{-24}\) Positive simple generators: \(\displaystyle 2g_{28}+2g_{22}\), \(\displaystyle g_{24}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}}\oplus 6V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 7V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 9V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}-2\psi_{1}+2\psi_{2}+8\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{2}+4\psi_{3}}\oplus V_{-2\psi_{1}+4\psi_{2}+6\psi_{3}} \oplus V_{\omega_{2}+6\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{3}} \oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}+6\psi_{3}}\oplus V_{2\psi_{1}+2\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+2\psi_{2}-2\psi_{3}} \oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{3}} \oplus 3V_{0}\oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{2}-4\psi_{3}}\oplus V_{-2\psi_{1}-2\psi_{2}} \oplus V_{4\psi_{1}-2\psi_{2}-6\psi_{3}}\oplus V_{\omega_{2}-6\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}-8\psi_{3}} \oplus V_{2\psi_{1}-4\psi_{2}-6\psi_{3}}\) Made total 9595847 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1+A^{2}_1\) ↪ \(E^{1}_6\) 28 out of 119
Subalgebra type: \(\displaystyle A^{4}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 1, 1, -1, 0), (1, 0, 0, 2, 0, 1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 4, 6, 4, 2): 8, \(\displaystyle A^{2}_1\): (1, 0, 2, 2, 2, 1): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-23}+g_{-27}\), \(\displaystyle g_{-18}+g_{-21}\) Positive simple generators: \(\displaystyle 2g_{27}+2g_{23}\), \(\displaystyle g_{21}+g_{18}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus 4V_{2\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{2}}\oplus 5V_{2\omega_{1}} \oplus 4V_{\omega_{2}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi_{1}+8\psi_{2}}\oplus V_{\omega_{2}+6\psi_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}+2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{1}+\omega_{2}+4\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{2}+6\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}} \oplus 2V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{1}-4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}-2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{2}-6\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}-8\psi_{2}}\oplus V_{\omega_{2}-6\psi_{1}-6\psi_{2}}\) Made total 51475 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{4}_1\) ↪ \(E^{1}_6\) 29 out of 119
Subalgebra type: \(\displaystyle 2A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1\) . Centralizer: \(\displaystyle A^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{4}_1+A^{1}_1\) , \(\displaystyle 3A^{4}_1\) , \(\displaystyle A^{1}_2+2A^{4}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 4, 6, 4, 2): 8, \(\displaystyle A^{4}_1\): (2, 0, 2, 2, 2, 2): 8 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-25}+g_{-26}\), \(\displaystyle g_{-7}+g_{-16}\) Positive simple generators: \(\displaystyle 2g_{26}+2g_{25}\), \(\displaystyle 2g_{16}+2g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0\\
0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0\\
0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{2}}\oplus 6V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}+2\psi_{1}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{1}+2\psi_{2}} \oplus V_{4\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{1}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{2}} \oplus 2V_{0}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 32934469 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_1+A^{1}_1\) ↪ \(E^{1}_6\) 30 out of 119
Subalgebra type: \(\displaystyle A^{5}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{5}_1\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, -1, -1, 0), (1, 0, 0, 2, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_1\): (3, 4, 5, 7, 5, 3): 10, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-22}+g_{-24}+g_{-28}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle 2g_{28}+g_{24}+2g_{22}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}} \oplus 4V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}+8\psi_{2}} \oplus V_{3\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{3\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{2}-6\psi_{1}+6\psi_{2}}\oplus V_{3\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+6\psi_{1}-6\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}+4\psi_{1}-8\psi_{2}}\oplus V_{\omega_{1}-6\psi_{2}}\) Made total 719 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_1+A^{4}_1\) ↪ \(E^{1}_6\) 31 out of 119
Subalgebra type: \(\displaystyle A^{5}_1+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{5}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, 1, 1, -1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_1\): (3, 4, 5, 7, 5, 3): 10, \(\displaystyle A^{4}_1\): (0, 0, 2, 2, 2, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-22}+g_{-24}+g_{-28}\), \(\displaystyle g_{-3}+g_{-10}\) Positive simple generators: \(\displaystyle 2g_{28}+g_{24}+2g_{22}\), \(\displaystyle 2g_{10}+2g_{3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5 & 0\\
0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10 & 0\\
0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{3\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{2}}\oplus 2V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{\omega_{1}+2\omega_{2}} \oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\omega_{2}+4\psi}\oplus V_{\omega_{1}+6\psi}\oplus V_{3\omega_{1}+2\omega_{2}+2\psi}\oplus V_{\omega_{1}+2\omega_{2}+2\psi} \oplus V_{4\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{3\omega_{1}+2\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}+2\omega_{2}-2\psi} \oplus V_{0}\oplus V_{2\omega_{1}+2\omega_{2}-4\psi}\oplus V_{\omega_{1}-6\psi}\) Made total 34939 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{6}_1\) ↪ \(E^{1}_6\) 32 out of 119
Subalgebra type: \(\displaystyle 2A^{6}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{6}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{6}_1\): (3, 4, 6, 8, 6, 3): 12, \(\displaystyle A^{6}_1\): (1, 2, 0, 2, 0, 1): 12 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-22}+g_{-23}+g_{-24}+g_{-25}\), \(\displaystyle g_{-1}-2g_{-2}-g_{-4}-g_{-6}\) Positive simple generators: \(\displaystyle g_{25}+2g_{24}+2g_{23}+g_{22}\), \(\displaystyle -g_{6}-2g_{4}-g_{2}+g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/3 & 0\\
0 & 1/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}12 & 0\\
0 & 12\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+3\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+3\omega_{2}+6\psi}\oplus V_{3\omega_{1}+\omega_{2}+6\psi}\oplus V_{2\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+2\omega_{2}} \oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+3\omega_{2}-6\psi}\oplus V_{3\omega_{1}+\omega_{2}-6\psi}\) Made total 6418412 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{1}_1\) ↪ \(E^{1}_6\) 33 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) . Centralizer: \(\displaystyle A^{3}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_2+A^{1}_1\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, -1, 0, -1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 4, 6, 8, 6, 4): 16, \(\displaystyle A^{1}_1\): (0, 1, 1, 2, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-12}+g_{-16}+g_{-22}+g_{-25}\), \(\displaystyle g_{-23}\) Positive simple generators: \(\displaystyle 2g_{25}+2g_{22}+2g_{16}+2g_{12}\), \(\displaystyle g_{23}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{4\omega_{1}+\omega_{2}}\oplus 4V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 4V_{2\omega_{1}} \oplus 4V_{\omega_{2}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi}\oplus V_{\omega_{2}+6\psi}\oplus V_{4\omega_{1}+\omega_{2}+2\psi}\oplus V_{2\omega_{1}+4\psi}\oplus V_{2\omega_{1}+\omega_{2}+2\psi} \oplus V_{4\psi}\oplus 2V_{4\omega_{1}}\oplus V_{\omega_{2}+2\psi}\oplus V_{4\omega_{1}+\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}} \oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}\oplus V_{4\omega_{1}-4\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{2\omega_{1}-4\psi}\oplus V_{-4\psi} \oplus V_{\omega_{2}-6\psi}\) Made total 3649565 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1\) ↪ \(E^{1}_6\) 34 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 4, 6, 8, 6, 4): 16, \(\displaystyle A^{3}_1\): (0, 2, 2, 3, 2, 0): 6 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}\), \(\displaystyle g_{-13}+g_{-14}-g_{-15}\) Positive simple generators: \(\displaystyle 2g_{21}+2g_{20}+2g_{18}+2g_{17}\), \(\displaystyle -g_{15}+g_{14}+g_{13}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\
0 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\
0 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{4\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{3\omega_{2}} \oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{2}+2\psi}\oplus V_{2\omega_{1}+\omega_{2}+2\psi} \oplus V_{4\psi}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{4\omega_{1}+\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{3\omega_{2}-2\psi}\oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\) Made total 5056886 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{4}_1\) ↪ \(E^{1}_6\) 35 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 4, 6, 8, 6, 4): 16, \(\displaystyle A^{4}_1\): (0, 2, 2, 4, 2, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}\), \(\displaystyle g_{-4}+g_{-19}\) Positive simple generators: \(\displaystyle 2g_{21}+2g_{20}+2g_{18}+2g_{17}\), \(\displaystyle 2g_{19}+2g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\
0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\
0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{4\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{2}}\oplus 2V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{4\omega_{1}}\oplus 3V_{2\omega_{2}} \oplus 2V_{2\omega_{1}}\) Made total 29361990 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1+A^{3}_1\) ↪ \(E^{1}_6\) 36 out of 119
Subalgebra type: \(\displaystyle A^{9}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (4, 5, 7, 10, 7, 4): 18, \(\displaystyle A^{3}_1\): (0, 1, 1, 0, 1, 0): 6 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}+g_{-23}\), \(\displaystyle g_{-2}-g_{-3}+g_{-5}\) Positive simple generators: \(\displaystyle g_{23}+2g_{21}+2g_{20}+2g_{18}+2g_{17}\), \(\displaystyle g_{5}-g_{3}+g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0\\
0 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0\\
0 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\) Made total 6484741 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{1}_1\) ↪ \(E^{1}_6\) 37 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2) Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (4, 6, 7, 10, 7, 4): 20, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-13}+g_{-14}+g_{-24}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle 4g_{24}+3g_{14}+3g_{13}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{4\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+2\psi_{1}+6\psi_{2}}\oplus V_{3\omega_{1}+\omega_{2}+6\psi_{2}}\oplus V_{3\omega_{1}-2\psi_{1}+6\psi_{2}} \oplus V_{4\omega_{1}+\omega_{2}+2\psi_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{\omega_{2}+2\psi_{1}} \oplus V_{4\omega_{1}+\omega_{2}-2\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{2}-2\psi_{1}}\oplus V_{3\omega_{1}+2\psi_{1}-6\psi_{2}} \oplus V_{3\omega_{1}+\omega_{2}-6\psi_{2}}\oplus V_{-4\psi_{1}}\oplus V_{3\omega_{1}-2\psi_{1}-6\psi_{2}}\) Made total 23596 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{2}_1\) ↪ \(E^{1}_6\) 38 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (4, 6, 7, 10, 7, 4): 20, \(\displaystyle A^{2}_1\): (0, 0, 1, 2, 1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-8}+g_{-19}+g_{-24}\), \(\displaystyle -g_{-4}+g_{-15}\) Positive simple generators: \(\displaystyle 4g_{24}+3g_{19}+3g_{8}\), \(\displaystyle g_{15}-g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{6\omega_{1}}\oplus 4V_{3\omega_{1}+\omega_{2}}\oplus 2V_{4\omega_{1}}\oplus 3V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{4\omega_{1}+2\omega_{2}} \oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{2}+4\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-4\psi_{1}+2\psi_{2}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{2\omega_{2}-4\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{3\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}}\) Made total 262477 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{10}_1\) ↪ \(E^{1}_6\) 39 out of 119
Subalgebra type: \(\displaystyle 2A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (4, 6, 7, 10, 7, 4): 20, \(\displaystyle A^{10}_1\): (0, 0, 3, 4, 3, 0): 20 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-13}+g_{-14}+g_{-24}\), \(\displaystyle g_{-3}+g_{-4}+g_{-5}\) Positive simple generators: \(\displaystyle 4g_{24}+3g_{14}+3g_{13}\), \(\displaystyle 3g_{5}+4g_{4}+3g_{3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\
0 & 1/5\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\
0 & 20\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+4\omega_{2}}\oplus V_{6\omega_{2}}\oplus 2V_{3\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+3\omega_{2}+6\psi}\oplus V_{4\omega_{1}+4\omega_{2}}\oplus V_{6\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{3\omega_{1}+3\omega_{2}-6\psi}\) Made total 1456960 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1+A^{1}_1\) ↪ \(E^{1}_6\) 40 out of 119
Subalgebra type: \(\displaystyle A^{11}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (4, 6, 8, 11, 8, 4): 22, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-13}+g_{-14}+g_{-15}+g_{-24}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle 4g_{24}+g_{15}+3g_{14}+3g_{13}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus 3V_{3\omega_{1}+\omega_{2}}\oplus 3V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 4V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+\omega_{2}+6\psi}\oplus V_{4\omega_{1}+6\psi}\oplus V_{2\omega_{1}+6\psi}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}} \oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{0} \oplus V_{3\omega_{1}+\omega_{2}-6\psi}\oplus V_{4\omega_{1}-6\psi}\oplus V_{2\omega_{1}-6\psi}\) Made total 956 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{20}_1+A^{1}_1\) ↪ \(E^{1}_6\) 41 out of 119
Subalgebra type: \(\displaystyle A^{20}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{20}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 3, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{20}_1\): (6, 8, 10, 14, 10, 6): 40, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-7}+g_{-13}+g_{-14}+g_{-16}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle 6g_{16}+4g_{14}+4g_{13}+6g_{7}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/10 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}40 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{8\omega_{1}}\oplus 2V_{6\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+12\psi}\oplus V_{6\omega_{1}+\omega_{2}+6\psi}\oplus V_{2\omega_{1}+\omega_{2}+6\psi}\oplus V_{8\omega_{1}} \oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{6\omega_{1}+\omega_{2}-6\psi}\oplus V_{0}\oplus V_{2\omega_{1}+\omega_{2}-6\psi} \oplus V_{4\omega_{1}-12\psi}\) Made total 2112126 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{28}_1+A^{2}_1\) ↪ \(E^{1}_6\) 42 out of 119
Subalgebra type: \(\displaystyle A^{28}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{28}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{28}_1\): (6, 10, 12, 18, 12, 6): 56, \(\displaystyle A^{2}_1\): (1, 0, 1, 0, 1, 1): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-2}+g_{-12}+g_{-15}+g_{-16}\), \(\displaystyle g_{-7}+g_{-11}\) Positive simple generators: \(\displaystyle 6g_{16}+6g_{15}+6g_{12}+10g_{2}\), \(\displaystyle g_{11}+g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{6\omega_{1}+2\omega_{2}}\oplus 2V_{6\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus 2V_{\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}+\omega_{2}+6\psi}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}+2\omega_{2}}\oplus V_{\omega_{2}+6\psi}\oplus V_{6\omega_{1}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{6\omega_{1}+\omega_{2}-6\psi}\oplus V_{0}\oplus V_{\omega_{2}-6\psi}\) Made total 1471223 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{28}_1+A^{8}_1\) ↪ \(E^{1}_6\) 43 out of 119
Subalgebra type: \(\displaystyle A^{28}_1+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{28}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{28}_1\): (6, 10, 12, 18, 12, 6): 56, \(\displaystyle A^{8}_1\): (2, 0, 2, 0, 2, 2): 16 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-2}+g_{-12}+g_{-15}+g_{-16}\), \(\displaystyle g_{-1}+g_{-3}+g_{-5}+g_{-6}\) Positive simple generators: \(\displaystyle 6g_{16}+6g_{15}+6g_{12}+10g_{2}\), \(\displaystyle 2g_{6}+2g_{5}+2g_{3}+2g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14 & 0\\
0 & 1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56 & 0\\
0 & 16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}+4\omega_{2}}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 6394164 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1+A^{1}_1\) ↪ \(E^{1}_6\) 44 out of 119
Subalgebra type: \(\displaystyle A^{35}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{35}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}}\oplus 9V_{\omega_{2}}\oplus 9V_{\omega_{1}}\oplus 16V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi_{1}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{3}+2\psi_{4}}\oplus V_{2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{2}-2\psi_{3}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+2\psi_{3}} \oplus V_{\omega_{2}+2\psi_{1}+2\psi_{3}-2\psi_{4}}\oplus V_{-2\psi_{3}+4\psi_{4}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{4\psi_{3}-2\psi_{4}}\oplus V_{\omega_{2}-2\psi_{2}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}+2\psi_{4}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}-2\psi_{4}} \oplus V_{\omega_{1}+2\psi_{2}-2\psi_{4}}\oplus 4V_{0}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{3}+2\psi_{4}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}-2\psi_{3}} \oplus V_{\omega_{2}-2\psi_{2}+2\psi_{3}-2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{4}}\oplus V_{-4\psi_{3}+2\psi_{4}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{2\psi_{3}-4\psi_{4}}\oplus V_{\omega_{2}-2\psi_{2}-2\psi_{3}} \oplus V_{\omega_{1}-2\psi_{1}-2\psi_{4}}\oplus V_{-2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{3}-2\psi_{4}}\) Made total 365 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2\) ↪ \(E^{1}_6\) 46 out of 119
Subalgebra type: \(\displaystyle B^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) . Centralizer: \(\displaystyle B^{1}_2\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\) Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1\) , \(\displaystyle B^{1}_2+A^{10}_1\) , \(\displaystyle B^{1}_2+2A^{1}_1\) , \(\displaystyle 2B^{1}_2\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 10. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{19}+g_{8}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-8}+g_{-19}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2\\
-2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 8V_{\omega_{2}}\oplus 5V_{\omega_{1}}\oplus 11V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{4\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}+2\psi_{2}+2\psi_{3}} \oplus V_{\omega_{1}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{2}+4\psi_{3}} \oplus V_{\omega_{1}+2\psi_{1}}\oplus V_{2\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{\omega_{1}}\oplus V_{\omega_{2}+2\psi_{1}+2\psi_{2}-4\psi_{3}}\oplus 3V_{0}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{\omega_{2}+2\psi_{2}-4\psi_{3}} \oplus V_{-4\psi_{2}+2\psi_{3}}\oplus V_{-2\psi_{1}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{2}-2\psi_{3}} \oplus V_{-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{1}-4\psi_{2}+2\psi_{3}}\) Made total 2180130 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(G^{1}_2\) ↪ \(E^{1}_6\) 47 out of 119
Subalgebra type: \(\displaystyle G^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle A^{2}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle G^{1}_2+A^{2}_1\) , \(\displaystyle G^{1}_2+A^{8}_1\) , \(\displaystyle A^{2}_2+G^{1}_2\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (2, 3, 4, 6, 4, 2): 6, (-1, -1, -2, -3, -2, -1): 2 Dimension of subalgebra generated by predefined or computed generators: 14. Negative simple generators: \(\displaystyle g_{-23}+g_{-30}+g_{-34}\), \(\displaystyle g_{35}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}+g_{23}\), \(\displaystyle g_{-35}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1\\
-1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\
-3 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}}\oplus 8V_{\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}} \oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}} \oplus V_{\omega_{1}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}} \oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 1031 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2\) ↪ \(E^{1}_6\) 48 out of 119
Subalgebra type: \(\displaystyle A^{2}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle A^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_2+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{4}_1\) , \(\displaystyle A^{2}_2+A^{1}_2\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2\\
-1/2 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2\\
-2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 3V_{\omega_{2}}\oplus 3V_{\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+2\psi_{2}}\oplus V_{\omega_{2}+2\psi_{1}} \oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}} \oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{\omega_{2}-2\psi_{2}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 26508 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2\) ↪ \(E^{1}_6\) 49 out of 119
Subalgebra type: \(\displaystyle A^{2}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle G^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (0, 1, -1, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_2+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{3}_1\) , \(\displaystyle A^{2}_2+A^{4}_1\) , \(\displaystyle A^{2}_2+A^{28}_1\) , \(\displaystyle A^{2}_2+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{1}_2\) , \(\displaystyle A^{2}_2+G^{1}_2\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{25}+g_{16}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle g_{-16}+g_{-25}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2\\
-1/2 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2\\
-2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 8V_{\omega_{1}+\omega_{2}}\oplus 14V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{-2\psi_{1}+6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{2}} \oplus V_{-4\psi_{1}+6\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}} \oplus V_{2\psi_{1}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}} \oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{4\psi_{1}-6\psi_{2}}\oplus V_{2\psi_{1}-6\psi_{2}}\) Made total 12521464 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{2}_2\) ↪ \(E^{1}_6\) 50 out of 119
Subalgebra type: \(\displaystyle B^{2}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (1, 0, -1, 0, 1, -1) Contained up to conjugation as a direct summand of: \(\displaystyle B^{2}_2+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (-2, 0, -2, -2, -2, -2): 8 Dimension of subalgebra generated by predefined or computed generators: 10. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{16}+g_{7}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle 2g_{-7}+2g_{-16}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2\\
-1/2 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -4\\
-4 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 5V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+6\psi_{2}}\oplus V_{2\omega_{2}-2\psi_{1}+6\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{1}}\oplus V_{-4\psi_{1}+6\psi_{2}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{4\psi_{1}-6\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{1}-6\psi_{2}} \oplus V_{\omega_{1}-6\psi_{2}}\) Made total 9280966 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_2\) ↪ \(E^{1}_6\) 51 out of 119
Subalgebra type: \(\displaystyle A^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_2\): (2, 3, 4, 6, 4, 2): 6, (-1, 0, -2, -3, -2, -1): 6 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-23}+g_{-30}+g_{-34}\), \(\displaystyle (1/2\sqrt{-3}+1/2)g_{24}+(-1/2\sqrt{-3}+1/2)g_{15}+g_{4}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}+g_{23}\), \(\displaystyle g_{-4}+(1/2\sqrt{-3}+1/2)g_{-15}+(-1/2\sqrt{-3}+1/2)g_{-24}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\
-3 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{2}}\oplus V_{3\omega_{1}}\oplus 7V_{\omega_{1}+\omega_{2}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+4\psi_{1}-2\psi_{2}} \oplus V_{3\omega_{2}}\oplus V_{3\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}-4\psi_{1}+2\psi_{2}}\oplus 2V_{0} \oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}}\) Made total 212017903 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_2\) ↪ \(E^{1}_6\) 52 out of 119
Subalgebra type: \(\displaystyle A^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_2\): (2, 3, 4, 6, 4, 2): 6, (0, -1, -1, -3, -1, 0): 6 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-29}+g_{-30}+g_{-31}\), \(\displaystyle g_{10}+g_{9}+g_{8}\) Positive simple generators: \(\displaystyle g_{31}+g_{30}+g_{29}\), \(\displaystyle g_{-8}+g_{-9}+g_{-10}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\
-3 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 3V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}} \oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\) Made total 141240 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{3}_2\) ↪ \(E^{1}_6\) 53 out of 119
Subalgebra type: \(\displaystyle B^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{3}_2\): (2, 3, 4, 6, 4, 2): 6, (-1, -2, -3, -6, -3, -1): 12 Dimension of subalgebra generated by predefined or computed generators: 10. Negative simple generators: \(\displaystyle g_{-29}+g_{-30}+g_{-31}\), \(\displaystyle (1/2\sqrt{-1}-1/2)g_{16}+g_{15}+(1/2\sqrt{-1}+1/2)g_{12}+g_{8}\) Positive simple generators: \(\displaystyle g_{31}+g_{30}+g_{29}\), \(\displaystyle 2g_{-8}+(-\sqrt{-1}+1)g_{-12}+2g_{-15}+(-\sqrt{-1}-1)g_{-16}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 1/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -6\\
-6 & 12\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+6\psi}\oplus V_{\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}-6\psi}\) Made total 7974385 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(G^{3}_2\) ↪ \(E^{1}_6\) 54 out of 119
Subalgebra type: \(\displaystyle G^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{3}_2\): (4, 5, 7, 10, 7, 4): 18, (-2, -2, -3, -5, -3, -2): 6 Dimension of subalgebra generated by predefined or computed generators: 14. Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}+g_{-23}\), \(\displaystyle (-1/4\sqrt{-7}-3/4)g_{28}+g_{26}+(1/2\sqrt{-7}-1/2)g_{24}\) Positive simple generators: \(\displaystyle g_{23}+2g_{21}+2g_{20}+2g_{18}+2g_{17}\), \(\displaystyle (-1/4\sqrt{-7}-1/4)g_{-24}+g_{-26}+(1/4\sqrt{-7}-3/4)g_{-28}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & -9\\
-9 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\) Made total 16487567 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_2\) ↪ \(E^{1}_6\) 55 out of 119
Subalgebra type: \(\displaystyle A^{5}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{5}_1\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{5}_2+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_2\): (3, 4, 5, 7, 5, 3): 10, (-1, -2, 0, -2, -2, -2): 10 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-22}+g_{-24}+g_{-28}\), \(\displaystyle g_{11}+2g_{8}+g_{1}\) Positive simple generators: \(\displaystyle 2g_{28}+g_{24}+2g_{22}\), \(\displaystyle g_{-1}+g_{-8}+2g_{-11}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5 & -1/5\\
-1/5 & 2/5\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10 & -5\\
-5 & 10\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{3\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{2}+2\psi}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{3\omega_{2}-2\psi}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\) Made total 4669812 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_2\) ↪ \(E^{1}_6\) 56 out of 119
Subalgebra type: \(\displaystyle A^{9}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_2\): (4, 5, 7, 10, 7, 4): 18, (-2, -1, -2, -5, -2, -2): 18 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}+g_{-23}\), \(\displaystyle (\sqrt{-1}-1)g_{10}+(-\sqrt{-1}-1)g_{9}+g_{8}+2\sqrt{-1}g_{6}+2\sqrt{-1}g_{1}\) Positive simple generators: \(\displaystyle g_{23}+2g_{21}+2g_{20}+2g_{18}+2g_{17}\), \(\displaystyle -\sqrt{-1}g_{-1}-\sqrt{-1}g_{-6}+g_{-8}+(\sqrt{-1}-1)g_{-9}+(-\sqrt{-1}-1)g_{-10}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & -1/9\\
-1/9 & 2/9\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & -9\\
-9 & 18\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 24846702 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{1}_1\) ↪ \(E^{1}_6\) 57 out of 119
Subalgebra type: \(\displaystyle 3A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle 2A^{1}_1\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 3 vectors:
(0, 0, 0, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle 4A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (1, 2, 2, 3, 2, 1): 2, \(\displaystyle A^{1}_1\): (1, 0, 1, 1, 1, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{-24}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{24}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{3}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}} \oplus 5V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi_{1}}\oplus V_{\omega_{3}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{2}+4\psi_{3}} \oplus V_{4\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{3}+2\psi_{1}-4\psi_{2}+2\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi_{1}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}-4\psi_{2}+2\psi_{3}}\oplus 3V_{0}\oplus V_{\omega_{2}+\omega_{3}+2\psi_{2}-4\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}-2\psi_{3}} \oplus V_{\omega_{1}+\omega_{3}-2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{3}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{-4\psi_{1}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}-2\psi_{3}}\) Made total 460 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1+2A^{1}_1\) ↪ \(E^{1}_6\) 58 out of 119
Subalgebra type: \(\displaystyle A^{2}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1+A^{1}_1\) . Centralizer: \(\displaystyle A^{2}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_3\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{2}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 2, 3, 4, 3, 2): 4, \(\displaystyle A^{1}_1\): (0, 1, 1, 2, 1, 0): 2, \(\displaystyle A^{1}_1\): (0, 1, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{-23}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle g_{23}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus 3V_{\omega_{2}+\omega_{3}}\oplus 4V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 4V_{2\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}+4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-4\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}} \oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}}\) Made total 713 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{2}_1+A^{1}_1\) ↪ \(E^{1}_6\) 59 out of 119
Subalgebra type: \(\displaystyle 2A^{2}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{2}_1\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2) Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{2}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 2, 3, 4, 3, 2): 4, \(\displaystyle A^{2}_1\): (0, 2, 1, 2, 1, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{-13}+g_{-14}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{14}+g_{13}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{2\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus 2V_{2\omega_{1}+\omega_{3}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}+6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+6\psi_{2}} \oplus V_{2\omega_{2}+\omega_{3}+2\psi_{1}}\oplus V_{2\omega_{1}+\omega_{3}+2\psi_{1}}\oplus V_{4\psi_{1}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{2}+\omega_{3}-2\psi_{1}}\oplus V_{2\omega_{1}+\omega_{3}-2\psi_{1}} \oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi_{2}}\oplus V_{-4\psi_{1}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-6\psi_{2}}\) Made total 679007 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{2}_1\) ↪ \(E^{1}_6\) 60 out of 119
Subalgebra type: \(\displaystyle 3A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{2}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 2, 3, 4, 3, 2): 4, \(\displaystyle A^{2}_1\): (0, 2, 1, 2, 1, 0): 4, \(\displaystyle A^{2}_1\): (0, 0, 1, 2, 1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{-8}+g_{-19}\), \(\displaystyle g_{-9}+g_{-10}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle g_{19}+g_{8}\), \(\displaystyle g_{10}+g_{9}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}+\omega_{3}} \oplus 2V_{2\omega_{3}}\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{2\omega_{3}}\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\) Made total 73384 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+A^{2}_1+A^{1}_1\) ↪ \(E^{1}_6\) 61 out of 119
Subalgebra type: \(\displaystyle A^{3}_1+A^{2}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1+A^{2}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 3, 4, 6, 4, 2): 6, \(\displaystyle A^{2}_1\): (1, 0, 1, 0, 1, 1): 4, \(\displaystyle A^{1}_1\): (0, 1, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-29}+g_{-30}+g_{-31}\), \(\displaystyle g_{-7}-g_{-11}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle g_{31}+g_{30}+g_{29}\), \(\displaystyle -g_{11}+g_{7}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{2}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}} \oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{2}} \oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{1}+\omega_{2}+6\psi}\oplus V_{\omega_{2}+6\psi}\oplus V_{\omega_{1}+2\omega_{2}+\omega_{3}} \oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}} \oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\oplus V_{2\omega_{1}+\omega_{2}-6\psi}\oplus V_{\omega_{2}-6\psi}\) Made total 743 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1+2A^{1}_1\) ↪ \(E^{1}_6\) 62 out of 119
Subalgebra type: \(\displaystyle A^{4}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1+A^{1}_1\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, -1, -1, 0), (1, 0, 0, 2, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 4, 6, 4, 2): 8, \(\displaystyle A^{1}_1\): (1, 0, 1, 1, 1, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-22}+g_{-28}\), \(\displaystyle g_{-24}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle 2g_{28}+2g_{22}\), \(\displaystyle g_{24}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{3}}\oplus 2V_{2\omega_{1}+\omega_{2}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}+8\psi_{2}} \oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+4\psi_{1}-2\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{3}-6\psi_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+6\psi_{1}-6\psi_{2}} \oplus 2V_{0}\oplus V_{2\omega_{1}+\omega_{3}-2\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+4\psi_{1}-8\psi_{2}}\oplus V_{\omega_{2}-6\psi_{2}}\) Made total 715 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{4}_1+A^{1}_1\) ↪ \(E^{1}_6\) 63 out of 119
Subalgebra type: \(\displaystyle 2A^{4}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{4}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1, 1, -1, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 4, 6, 4, 2): 8, \(\displaystyle A^{4}_1\): (2, 0, 2, 2, 2, 2): 8, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-25}+g_{-26}\), \(\displaystyle g_{-7}+g_{-16}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle 2g_{26}+2g_{25}\), \(\displaystyle 2g_{16}+2g_{7}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0 & 0\\
0 & 1/2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0 & 0\\
0 & 8 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{1}+2\omega_{2}+\omega_{3}}\oplus V_{4\omega_{2}}\oplus 2V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\omega_{2}+4\psi}\oplus V_{\omega_{3}+6\psi}\oplus V_{2\omega_{1}+2\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\omega_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}+\omega_{3}-2\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{0}\oplus V_{2\omega_{1}+2\omega_{2}-4\psi}\oplus V_{\omega_{3}-6\psi}\) Made total 954 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{4}_1\) ↪ \(E^{1}_6\) 64 out of 119
Subalgebra type: \(\displaystyle 3A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{4}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 4, 6, 4, 2): 8, \(\displaystyle A^{4}_1\): (2, 0, 2, 2, 2, 2): 8, \(\displaystyle A^{4}_1\): (0, 0, 2, 2, 2, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-25}+g_{-26}\), \(\displaystyle g_{-7}+g_{-16}\), \(\displaystyle g_{-5}+g_{-9}\) Positive simple generators: \(\displaystyle 2g_{26}+2g_{25}\), \(\displaystyle 2g_{16}+2g_{7}\), \(\displaystyle 2g_{9}+2g_{5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0 & 0\\
0 & 1/2 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0 & 0\\
0 & 8 & 0\\
0 & 0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus V_{4\omega_{3}}\oplus V_{4\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\) Made total 45630 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1+A^{1}_1\) ↪ \(E^{1}_6\) 65 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{3}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 4, 6, 8, 6, 4): 16, \(\displaystyle A^{3}_1\): (0, 2, 2, 3, 2, 0): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}\), \(\displaystyle g_{-13}+g_{-14}-g_{-15}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle 2g_{21}+2g_{20}+2g_{18}+2g_{17}\), \(\displaystyle -g_{15}+g_{14}+g_{13}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0\\
0 & 2/3 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0\\
0 & 6 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{2}+\omega_{3}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}} \oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 925 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+2A^{1}_1\) ↪ \(E^{1}_6\) 66 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1+A^{1}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (4, 6, 7, 10, 7, 4): 20, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-13}+g_{-14}+g_{-24}\), \(\displaystyle g_{-15}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle 4g_{24}+3g_{14}+3g_{13}\), \(\displaystyle g_{15}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{3}}\oplus 2V_{3\omega_{1}+\omega_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+\omega_{3}+6\psi}\oplus V_{3\omega_{1}+\omega_{2}+6\psi}\oplus V_{4\omega_{1}+\omega_{2}+\omega_{3}} \oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0} \oplus V_{3\omega_{1}+\omega_{3}-6\psi}\oplus V_{3\omega_{1}+\omega_{2}-6\psi}\) Made total 638 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{1}_1\) ↪ \(E^{1}_6\) 67 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle A^{1}_2\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_2\) Basis of Cartan of centralizer: 3 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_2+2A^{1}_1\) , \(\displaystyle A^{1}_2+A^{4}_1+A^{1}_1\) , \(\displaystyle 2A^{1}_2+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1, 1): 2 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-11}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{11}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus 3V_{\omega_{2}+\omega_{3}}\oplus 3V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{3}} \oplus 3V_{\omega_{2}}\oplus 3V_{\omega_{1}}\oplus 9V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+6\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}+4\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{2}+2\psi_{3}} \oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}+2\psi_{3}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}+2\psi_{1}-2\psi_{3}} \oplus 3V_{0}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{2}-4\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}} \oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-4\psi_{3}} \oplus V_{-2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{3}}\oplus V_{\omega_{3}-6\psi_{3}}\) Made total 450 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{2}_1\) ↪ \(E^{1}_6\) 68 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, -1, 0, 0, 0), (0, 0, 0, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{2}_1\): (1, 0, 1, 0, 1, 1): 4 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-7}+g_{-11}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{11}+g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}}\oplus 2V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}} \oplus V_{\omega_{1}+\omega_{2}}\oplus 4V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+\omega_{3}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+6\psi_{2}} \oplus V_{\omega_{3}+6\psi_{1}}\oplus V_{\omega_{1}+2\omega_{3}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+2\omega_{3}+2\psi_{1}-2\psi_{2}} \oplus 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}-4\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-4\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{2}+\omega_{3}-4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{3}-6\psi_{1}}\oplus V_{\omega_{3}-6\psi_{2}}\) Made total 2784 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{4}_1\) ↪ \(E^{1}_6\) 69 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle A^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_2+A^{4}_1+A^{1}_1\) , \(\displaystyle A^{1}_2+2A^{4}_1\) , \(\displaystyle 2A^{1}_2+A^{4}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{4}_1\): (0, 0, 0, 0, 2, 2): 8 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-5}+g_{-6}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle 2g_{6}+2g_{5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{3}}\oplus 3V_{\omega_{2}+2\omega_{3}}\oplus 3V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\omega_{3}+2\psi_{2}}\oplus V_{\omega_{2}+2\omega_{3}+2\psi_{1}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{4\omega_{3}} \oplus V_{\omega_{2}+2\omega_{3}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\omega_{3}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+2\omega_{3}-2\psi_{1}}\oplus V_{\omega_{2}+2\omega_{3}-2\psi_{2}} \oplus 2V_{0}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 2780 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{5}_1\) ↪ \(E^{1}_6\) 70 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+A^{5}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{5}_1\): (1, 0, 1, 0, 2, 2): 10 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-5}+g_{-6}+g_{-7}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{7}+2g_{6}+2g_{5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2/5\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 10\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{3}}\oplus V_{\omega_{2}+3\omega_{3}}\oplus V_{\omega_{1}+3\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}} \oplus 2V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{3}} \oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+6\psi}\oplus V_{\omega_{1}+2\omega_{3}+4\psi}\oplus V_{\omega_{2}+3\omega_{3}+2\psi}\oplus V_{\omega_{2}+\omega_{3}+2\psi} \oplus V_{4\omega_{3}}\oplus 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+3\omega_{3}-2\psi}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi} \oplus V_{\omega_{2}+2\omega_{3}-4\psi}\oplus V_{\omega_{3}-6\psi}\) Made total 10697 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{8}_1\) ↪ \(E^{1}_6\) 71 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{8}_1\): (2, 0, 2, 0, 2, 2): 16 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-1}+g_{-3}+g_{-5}+g_{-6}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle 2g_{6}+2g_{5}+2g_{3}+2g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+4\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus 2V_{4\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}} \oplus 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\) Made total 30250 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{1}_1\) ↪ \(E^{1}_6\) 72 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2) Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 13. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{14}+g_{13}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-13}+g_{-14}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{3}}\oplus 4V_{\omega_{2}} \oplus V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi_{1}+6\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}+6\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}+6\psi_{2}} \oplus V_{4\psi_{1}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}}\oplus V_{\omega_{3}+2\psi_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}}\oplus V_{\omega_{3}-2\psi_{1}}\oplus V_{\omega_{2}+2\psi_{1}-6\psi_{2}} \oplus V_{-4\psi_{1}}\oplus V_{\omega_{2}+\omega_{3}-6\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}-6\psi_{2}}\) Made total 22148 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{2}_1\) ↪ \(E^{1}_6\) 73 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \(\displaystyle A^{2}_1\): (0, 0, 1, 2, 1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 13. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{19}+g_{8}\), \(\displaystyle -g_{-4}+g_{-15}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-8}+g_{-19}\), \(\displaystyle g_{15}-g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{3}}\oplus 3V_{2\omega_{3}}\oplus 4V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{3}+4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{3}-4\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{2}+\omega_{3}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}-2\psi_{2}}\) Made total 262431 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{10}_1\) ↪ \(E^{1}_6\) 74 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \(\displaystyle A^{10}_1\): (0, 0, 3, 4, 3, 0): 20 Dimension of subalgebra generated by predefined or computed generators: 13. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{14}+g_{13}\), \(\displaystyle g_{-3}+g_{-4}+g_{-5}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-13}+g_{-14}\), \(\displaystyle 3g_{5}+4g_{4}+3g_{3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 1/5\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 20\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus 2V_{\omega_{2}+3\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+3\omega_{3}+6\psi}\oplus V_{6\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{0}\oplus V_{\omega_{2}+3\omega_{3}-6\psi}\) Made total 1458590 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(G^{1}_2+A^{2}_1\) ↪ \(E^{1}_6\) 75 out of 119
Subalgebra type: \(\displaystyle G^{1}_2+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle G^{1}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (2, 3, 4, 6, 4, 2): 6, (-1, -1, -2, -3, -2, -1): 2, \(\displaystyle A^{2}_1\): (1, 0, 1, 0, 1, 1): 4 Dimension of subalgebra generated by predefined or computed generators: 17. Negative simple generators: \(\displaystyle g_{-29}+g_{-30}+g_{-31}\), \(\displaystyle g_{35}\), \(\displaystyle g_{-7}-g_{-11}\) Positive simple generators: \(\displaystyle g_{31}+g_{30}+g_{29}\), \(\displaystyle g_{-35}\), \(\displaystyle -g_{11}+g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3 & 0\\
-3 & 2 & 0\\
0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}} \oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{3}+6\psi}\oplus V_{\omega_{3}+6\psi}\oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}} \oplus V_{\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{3}-6\psi}\oplus V_{\omega_{3}-6\psi}\) Made total 1471286 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(G^{1}_2+A^{8}_1\) ↪ \(E^{1}_6\) 76 out of 119
Subalgebra type: \(\displaystyle G^{1}_2+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle G^{1}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (2, 3, 4, 6, 4, 2): 6, (-1, -1, -2, -3, -2, -1): 2, \(\displaystyle A^{8}_1\): (2, 0, 2, 0, 2, 2): 16 Dimension of subalgebra generated by predefined or computed generators: 17. Negative simple generators: \(\displaystyle g_{-29}+g_{-30}+g_{-31}\), \(\displaystyle g_{35}\), \(\displaystyle g_{-1}+g_{-3}-g_{-5}+g_{-6}\) Positive simple generators: \(\displaystyle g_{31}+g_{30}+g_{29}\), \(\displaystyle g_{-35}\), \(\displaystyle 2g_{6}-2g_{5}+2g_{3}+2g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3 & 0\\
-3 & 2 & 0\\
0 & 0 & 16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+4\omega_{3}}\oplus V_{4\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}}\) Made total 5318551 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{1}_1\) ↪ \(E^{1}_6\) 77 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, -1, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, \(\displaystyle A^{1}_1\): (0, 1, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle g_{-8}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}+\omega_{3}}\oplus V_{2\omega_{1}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+6\psi}\oplus V_{2\omega_{2}+4\psi}\oplus V_{\omega_{1}+4\psi}\oplus V_{2\omega_{1}+\omega_{3}+2\psi}\oplus V_{\omega_{2}+\omega_{3}+2\psi} \oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}+\omega_{3}-2\psi}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi} \oplus V_{2\omega_{1}-4\psi}\oplus V_{\omega_{2}-4\psi}\oplus V_{\omega_{3}-6\psi}\) Made total 630 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{1}_1\) ↪ \(E^{1}_6\) 78 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: \(\displaystyle A^{3}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_2+A^{1}_1\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, -1, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_2+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \(\displaystyle A^{1}_1\): (0, 1, 1, 2, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{25}+g_{16}\), \(\displaystyle g_{-23}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle g_{-16}+g_{-25}\), \(\displaystyle g_{23}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 4V_{\omega_{3}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+6\psi}\oplus V_{\omega_{1}+\omega_{2}+4\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\psi} \oplus V_{\omega_{3}+2\psi}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi} \oplus V_{0}\oplus V_{\omega_{3}-2\psi}\oplus V_{\omega_{1}+\omega_{2}-4\psi}\oplus V_{-4\psi}\oplus V_{\omega_{3}-6\psi}\) Made total 958 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{3}_1\) ↪ \(E^{1}_6\) 79 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_2+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \(\displaystyle A^{3}_1\): (0, 2, 2, 3, 2, 0): 6 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{-13}+g_{-14}-g_{-15}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle -g_{15}+g_{14}+g_{13}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}+2\omega_{3}}\oplus 2V_{3\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}} \oplus V_{\omega_{1}+\omega_{2}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{3}+2\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\psi}\oplus V_{\omega_{1}+\omega_{2}+2\omega_{3}} \oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{3\omega_{3}-2\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi} \oplus V_{0}\oplus V_{-4\psi}\) Made total 5136524 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{4}_1\) ↪ \(E^{1}_6\) 80 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, \(\displaystyle A^{4}_1\): (0, 2, 0, 2, 0, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle g_{-2}+g_{-4}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle 2g_{4}+2g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{3}}\oplus V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}} \oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\) Made total 3810 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{4}_1\) ↪ \(E^{1}_6\) 81 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \(\displaystyle A^{4}_1\): (0, 2, 2, 4, 2, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-30}+g_{-34}\), \(\displaystyle g_{25}+g_{16}\), \(\displaystyle g_{-8}+g_{-9}-g_{-10}+g_{-19}\) Positive simple generators: \(\displaystyle g_{34}+g_{30}\), \(\displaystyle g_{-16}+g_{-25}\), \(\displaystyle g_{19}-g_{10}+g_{9}+g_{8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}+2\omega_{3}}\oplus 3V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\) Made total 6862360 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{28}_1\) ↪ \(E^{1}_6\) 82 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{28}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \(\displaystyle A^{28}_1\): (0, 6, 6, 10, 6, 0): 56 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{-2}-g_{-3}+g_{-4}+g_{-5}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle 6g_{5}+10g_{4}-6g_{3}+6g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 1/14\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 56\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}+6\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 6570789 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{2}_2+A^{1}_1\) ↪ \(E^{1}_6\) 83 out of 119
Subalgebra type: \(\displaystyle B^{2}_2+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{2}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 3, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (-2, 0, -2, -2, -2, -2): 8, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 13. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{16}+g_{7}\), \(\displaystyle g_{-15}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle 2g_{-7}+2g_{-16}\), \(\displaystyle g_{15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1/2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -4 & 0\\
-4 & 8 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+12\psi}\oplus V_{2\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0} \oplus V_{2\omega_{2}+\omega_{3}-6\psi}\oplus V_{\omega_{1}-12\psi}\) Made total 2168207 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_2+A^{1}_1\) ↪ \(E^{1}_6\) 84 out of 119
Subalgebra type: \(\displaystyle A^{5}_2+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{5}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_2\): (3, 4, 5, 7, 5, 3): 10, (-1, -2, 0, -2, -2, -2): 10, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-22}+g_{-24}+g_{-28}\), \(\displaystyle g_{11}+2g_{8}+g_{1}\), \(\displaystyle g_{-10}\) Positive simple generators: \(\displaystyle 2g_{28}+g_{24}+2g_{22}\), \(\displaystyle g_{-1}+g_{-8}+2g_{-11}\), \(\displaystyle g_{10}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5 & -1/5 & 0\\
-1/5 & 2/5 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10 & -5 & 0\\
-5 & 10 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{2}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 812 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_3\) ↪ \(E^{1}_6\) 85 out of 119
Subalgebra type: \(\displaystyle A^{1}_3\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle 2A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\) Basis of Cartan of centralizer: 3 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0), (0, 0, 0, 0, 0, 1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_3+A^{1}_1\) , \(\displaystyle A^{1}_3+A^{2}_1\) , \(\displaystyle A^{1}_3+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_3\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 15. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{3}}\oplus 4V_{\omega_{3}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 7V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+4\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}+2\psi_{3}}\oplus V_{\omega_{3}+2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{2}-2\psi_{3}}\oplus 3V_{0}\oplus V_{\omega_{1}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}-2\psi_{3}}\oplus V_{\omega_{3}-4\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{2}-4\psi_{3}}\) Made total 448 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_3\) ↪ \(E^{1}_6\) 86 out of 119
Subalgebra type: \(\displaystyle B^{1}_3\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle A^{2}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_3\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_3+A^{2}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_3\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, -1, -2, -1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 21. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{15}+g_{4}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}+g_{-15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -2\\
0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 4V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus 3V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+2\psi_{1}-4\psi_{2}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}-2\psi_{2}}\) Made total 201627 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3\) ↪ \(E^{1}_6\) 87 out of 119
Subalgebra type: \(\displaystyle C^{1}_3\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle C^{1}_3+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, -1, -1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 21. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle g_{15}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{-15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & -1\\
0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & -2\\
0 & -2 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{\omega_{1}+2\psi}\oplus V_{2\omega_{1}}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{3}-2\psi} \oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\) Made total 632 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_3\) ↪ \(E^{1}_6\) 88 out of 119
Subalgebra type: \(\displaystyle A^{2}_3\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_3+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_3\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, 0, -1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 15. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle -g_{5}+g_{3}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{-3}-g_{-5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & -1/2\\
0 & -1/2 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & -2\\
0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{2\omega_{3}+2\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}\oplus V_{2\omega_{3}-2\psi} \oplus V_{2\omega_{1}-2\psi}\oplus V_{-4\psi}\) Made total 8556 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(4A^{1}_1\) ↪ \(E^{1}_6\) 89 out of 119
Subalgebra type: \(\displaystyle 4A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle 3A^{1}_1\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (1, 2, 2, 3, 2, 1): 2, \(\displaystyle A^{1}_1\): (1, 0, 1, 1, 1, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 12. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{-24}\), \(\displaystyle g_{-15}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{24}\), \(\displaystyle g_{15}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus 2V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}} \oplus 2V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}} \oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+\omega_{4}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{4}-2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}+4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+\omega_{2}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{\omega_{3}+\omega_{4}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-4\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{4}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{2}+\omega_{3}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{2}+\omega_{4}-2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}-2\psi_{2}}\) Made total 543 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{2}_1+2A^{1}_1\) ↪ \(E^{1}_6\) 90 out of 119
Subalgebra type: \(\displaystyle 2A^{2}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{2}_1+A^{1}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 2, 3, 4, 3, 2): 4, \(\displaystyle A^{2}_1\): (0, 2, 1, 2, 1, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 12. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{-13}+g_{-14}\), \(\displaystyle g_{-15}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{14}+g_{13}\), \(\displaystyle g_{15}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0 & 0\\
0 & 4 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{\omega_{1}+\omega_{2}+\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{4}+6\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{2}+\omega_{3}+\omega_{4}} \oplus V_{2\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}+\omega_{4}-6\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\) Made total 731 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+2A^{1}_1\) ↪ \(E^{1}_6\) 91 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2+A^{1}_1\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, -1, 0, 0, 0), (0, 0, 0, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1, 1): 2, \(\displaystyle A^{1}_1\): (1, 0, 1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 14. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-11}\), \(\displaystyle g_{-7}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{11}\), \(\displaystyle g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}} \oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 2V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+\omega_{4}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+6\psi_{2}} \oplus V_{\omega_{4}+6\psi_{1}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{4}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}+\omega_{4}+2\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}-4\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-4\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{2}+\omega_{3}-4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{4}-2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{4}-6\psi_{1}} \oplus V_{\omega_{3}-6\psi_{2}}\) Made total 535 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{4}_1+A^{1}_1\) ↪ \(E^{1}_6\) 92 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+A^{4}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2+A^{4}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{4}_1\): (0, 0, 0, 0, 2, 2): 8, \(\displaystyle A^{1}_1\): (1, 0, 1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 14. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-5}+g_{-6}\), \(\displaystyle g_{-7}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle 2g_{6}+2g_{5}\), \(\displaystyle g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 1/2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 8 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+2\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+2\omega_{3}+\omega_{4}}\oplus V_{4\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}} \oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{4}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{4}+6\psi}\oplus V_{\omega_{1}+2\omega_{3}+4\psi}\oplus V_{\omega_{2}+2\omega_{3}+\omega_{4}+2\psi}\oplus V_{4\omega_{3}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+2\omega_{3}+\omega_{4}-2\psi}\oplus V_{0} \oplus V_{\omega_{2}+2\omega_{3}-4\psi}\oplus V_{\omega_{4}-6\psi}\) Made total 624 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+2A^{4}_1\) ↪ \(E^{1}_6\) 93 out of 119
Subalgebra type: \(\displaystyle A^{1}_2+2A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2+A^{4}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{4}_1\): (0, 0, 0, 0, 2, 2): 8, \(\displaystyle A^{4}_1\): (2, 0, 2, 0, 0, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 14. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-5}+g_{-6}\), \(\displaystyle g_{-1}+g_{-3}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle 2g_{6}+2g_{5}\), \(\displaystyle 2g_{3}+2g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 1/2 & 0\\
0 & 0 & 0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 8 & 0\\
0 & 0 & 0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+2\omega_{3}+2\omega_{4}}\oplus V_{\omega_{1}+2\omega_{3}+2\omega_{4}}\oplus V_{4\omega_{4}}\oplus V_{4\omega_{3}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 3798 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{1}_2\) ↪ \(E^{1}_6\) 94 out of 119
Subalgebra type: \(\displaystyle 2A^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2+A^{1}_1\) . Centralizer: \(\displaystyle A^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{1}_2+A^{1}_1\) , \(\displaystyle 2A^{1}_2+A^{4}_1\) , \(\displaystyle 3A^{1}_2\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{1}_2\): (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2 Dimension of subalgebra generated by predefined or computed generators: 16. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-11}\), \(\displaystyle g_{6}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{11}\), \(\displaystyle g_{-6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{3}+\omega_{4}}\oplus 3V_{\omega_{2}+\omega_{4}}\oplus 3V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{4}+2\psi_{1}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{2}+\omega_{4}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}}\oplus V_{\omega_{2}+\omega_{4}-2\psi_{2}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 533 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+2A^{1}_1\) ↪ \(E^{1}_6\) 95 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{1}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 1, 1, 1, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 16. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{14}+g_{13}\), \(\displaystyle g_{-15}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-13}+g_{-14}\), \(\displaystyle g_{15}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}} \oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+\omega_{4}+6\psi}\oplus V_{\omega_{2}+\omega_{3}+6\psi}\oplus V_{\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}} \oplus V_{\omega_{3}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-6\psi} \oplus V_{\omega_{2}+\omega_{3}-6\psi}\) Made total 628 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2B^{1}_2\) ↪ \(E^{1}_6\) 96 out of 119
Subalgebra type: \(\displaystyle 2B^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{1}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \(\displaystyle B^{1}_2\): (0, 0, 1, 1, 1, 0): 2, (0, 0, -1, 0, -1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 20. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{14}+g_{13}\), \(\displaystyle g_{-15}\), \(\displaystyle g_{5}+g_{3}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-13}+g_{-14}\), \(\displaystyle g_{15}\), \(\displaystyle g_{-3}+g_{-5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 1 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+\omega_{4}+6\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-6\psi}\) Made total 8550 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{3}_1+A^{1}_1\) ↪ \(E^{1}_6\) 97 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{3}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \(\displaystyle A^{3}_1\): (0, 2, 2, 3, 2, 0): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 14. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{-13}+g_{-14}-g_{-15}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle -g_{15}+g_{14}+g_{13}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2/3 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 6 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}+2\omega_{3}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 913 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{1}_2\) ↪ \(E^{1}_6\) 98 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{1}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \(\displaystyle A^{1}_2\): (0, 1, 1, 2, 1, 0): 2, (0, 0, 0, -1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 16. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{-23}\), \(\displaystyle g_{4}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle g_{23}\), \(\displaystyle g_{-4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{2}} \oplus V_{\omega_{4}}\oplus V_{\omega_{3}}\) Made total 130682 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{1}_2\) ↪ \(E^{1}_6\) 99 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{1}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, \(\displaystyle A^{1}_2\): (0, 1, 0, 1, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 16. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle g_{-8}\), \(\displaystyle g_{4}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{8}\), \(\displaystyle g_{-4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{3}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}} \oplus V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\) Made total 721 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+G^{1}_2\) ↪ \(E^{1}_6\) 100 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+G^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{3}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \(\displaystyle G^{1}_2\): (0, 2, 2, 3, 2, 0): 6, (0, -1, -1, -1, -1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 22. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{-13}+g_{-14}-g_{-15}\), \(\displaystyle g_{19}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle -g_{15}+g_{14}+g_{13}\), \(\displaystyle g_{-19}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2/3 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 6 & -3\\
0 & 0 & -3 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{4}}\) Made total 921 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_3+A^{1}_1\) ↪ \(E^{1}_6\) 101 out of 119
Subalgebra type: \(\displaystyle A^{1}_3+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 2, 0, -2, -1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_3+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_3\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 0, 1): 2 Dimension of subalgebra generated by predefined or computed generators: 18. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{-6}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}} \oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{4}+6\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}+8\psi_{2}}\oplus V_{\omega_{3}+2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{2}+\omega_{4}-2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{2}+\omega_{4}+2\psi_{1}-2\psi_{2}} \oplus V_{-4\psi_{1}+4\psi_{2}}\oplus 2V_{0}\oplus V_{4\psi_{1}-4\psi_{2}}\oplus V_{\omega_{3}+\omega_{4}-6\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{1}+2\psi_{1}-8\psi_{2}}\) Made total 531 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_3+A^{2}_1\) ↪ \(E^{1}_6\) 102 out of 119
Subalgebra type: \(\displaystyle A^{1}_3+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_3\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, \(\displaystyle A^{2}_1\): (1, 0, 0, 0, 0, 1): 4 Dimension of subalgebra generated by predefined or computed generators: 18. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{-1}+g_{-6}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{6}+g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+2\omega_{4}}\oplus 2V_{2\omega_{4}}\oplus 2V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+\omega_{4}+6\psi}\oplus V_{\omega_{1}+\omega_{4}+6\psi}\oplus V_{\omega_{2}+2\omega_{4}}\oplus 2V_{2\omega_{4}} \oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{3}+\omega_{4}-6\psi}\oplus V_{\omega_{1}+\omega_{4}-6\psi}\) Made total 3288 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_3+A^{2}_1\) ↪ \(E^{1}_6\) 103 out of 119
Subalgebra type: \(\displaystyle B^{1}_3+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_3\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_3\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, -1, -2, -1, 0): 4, \(\displaystyle A^{2}_1\): (0, 0, 1, 0, 1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 24. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{10}+g_{9}\), \(\displaystyle g_{-3}+g_{-5}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-9}+g_{-10}\), \(\displaystyle g_{5}+g_{3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -2 & 0\\
0 & -2 & 4 & 0\\
0 & 0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{4}}\oplus V_{2\omega_{4}}\oplus 2V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+\omega_{4}+6\psi}\oplus V_{\omega_{1}+2\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{3}+\omega_{4}-6\psi}\) Made total 1466930 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3+A^{1}_1\) ↪ \(E^{1}_6\) 104 out of 119
Subalgebra type: \(\displaystyle C^{1}_3+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, -1, -1, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 24. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle g_{15}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{-15}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & -2 & 0\\
0 & -2 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{2}}\) Made total 715 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_3+A^{1}_1\) ↪ \(E^{1}_6\) 105 out of 119
Subalgebra type: \(\displaystyle A^{2}_3+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_3\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, 0, -1, 0): 4, \(\displaystyle A^{1}_1\): (0, 1, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 18. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle -g_{5}+g_{3}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{-3}-g_{-5}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & -1/2 & 0\\
0 & -1/2 & 1 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & -2 & 0\\
0 & -2 & 4 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\) Made total 796 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_4\) ↪ \(E^{1}_6\) 106 out of 119
Subalgebra type: \(\displaystyle A^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, -1, -2) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_4+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_4\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 24. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+6\psi_{2}}\oplus V_{\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{4}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}} \oplus V_{\omega_{4}-6\psi_{2}}\) Made total 531 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(D^{1}_4\) ↪ \(E^{1}_6\) 107 out of 119
Subalgebra type: \(\displaystyle D^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle D^{1}_4\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, -1, -1, -1, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 28. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{15}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-15}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & -1\\
0 & -1 & 2 & 0\\
0 & -1 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & -1\\
0 & -1 & 2 & 0\\
0 & -1 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{4}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{4}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}-2\psi_{2}}\) Made total 535 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_4\) ↪ \(E^{1}_6\) 108 out of 119
Subalgebra type: \(\displaystyle B^{1}_4\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_4\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, -1, 0, -1, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 36. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}+g_{3}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-3}+g_{-5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{4}+6\psi}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\oplus V_{\omega_{4}-6\psi}\) Made total 3288 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_4\) ↪ \(E^{1}_6\) 109 out of 119
Subalgebra type: \(\displaystyle C^{1}_4\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_4\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, 0, -1, 0): 4, (0, 0, 0, -1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 36. Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle -g_{5}+g_{3}\), \(\displaystyle g_{4}\) Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{-3}-g_{-5}\), \(\displaystyle g_{-4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & -1/2 & 0\\
0 & -1/2 & 1 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & -2 & 0\\
0 & -2 & 4 & -2\\
0 & 0 & -2 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus V_{\omega_{4}}\) Made total 796 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(F^{1}_4\) ↪ \(E^{1}_6\) 110 out of 119
Subalgebra type: \(\displaystyle F^{1}_4\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle F^{1}_4\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, -1, -2, -1, 0): 4, (-1, 0, 0, 0, 0, -1): 4 Dimension of subalgebra generated by predefined or computed generators: 52. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{10}+g_{9}\), \(\displaystyle g_{6}+g_{1}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-9}+g_{-10}\), \(\displaystyle g_{-1}+g_{-6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 1 & -1/2\\
0 & 0 & -1/2 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -2 & 0\\
0 & -2 & 4 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{4}}\oplus V_{\omega_{1}}\) Made total 133227 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{1}_2+A^{1}_1\) ↪ \(E^{1}_6\) 111 out of 119
Subalgebra type: \(\displaystyle 2A^{1}_2+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle 2A^{1}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{1}_2\): (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, \(\displaystyle A^{1}_1\): (1, 0, 1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 19. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-11}\), \(\displaystyle g_{6}\), \(\displaystyle g_{-7}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{11}\), \(\displaystyle g_{-6}\), \(\displaystyle g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{5}}\oplus V_{2\omega_{5}}\oplus V_{\omega_{3}+\omega_{4}} \oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{5}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{5}+6\psi}\oplus V_{\omega_{1}+\omega_{3}+4\psi}\oplus V_{\omega_{2}+\omega_{4}+\omega_{5}+2\psi}\oplus V_{2\omega_{5}} \oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{5}-2\psi}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-4\psi} \oplus V_{\omega_{5}-6\psi}\) Made total 618 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{1}_2+A^{4}_1\) ↪ \(E^{1}_6\) 112 out of 119
Subalgebra type: \(\displaystyle 2A^{1}_2+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{1}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{1}_2\): (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, \(\displaystyle A^{4}_1\): (2, 0, 2, 0, 0, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 19. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-11}\), \(\displaystyle g_{6}\), \(\displaystyle g_{-1}+g_{-3}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{11}\), \(\displaystyle g_{-6}\), \(\displaystyle 2g_{3}+2g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{5}}\oplus V_{\omega_{2}+\omega_{4}+2\omega_{5}}\oplus V_{\omega_{1}+\omega_{3}+2\omega_{5}}\oplus V_{2\omega_{5}} \oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 3798 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_3+2A^{1}_1\) ↪ \(E^{1}_6\) 113 out of 119
Subalgebra type: \(\displaystyle A^{1}_3+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3+A^{1}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_3\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 0, 1): 2, \(\displaystyle A^{1}_1\): (1, 0, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 21. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{-6}\), \(\displaystyle g_{-1}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{6}\), \(\displaystyle g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{2\omega_{5}}\oplus V_{\omega_{3}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{5}} \oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+\omega_{5}+6\psi}\oplus V_{\omega_{1}+\omega_{4}+6\psi}\oplus V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{2\omega_{5}} \oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{5}-6\psi}\oplus V_{\omega_{3}+\omega_{4}-6\psi}\) Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_4+A^{1}_1\) ↪ \(E^{1}_6\) 114 out of 119
Subalgebra type: \(\displaystyle A^{1}_4+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_4\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -4)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_4\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, \(\displaystyle A^{1}_1\): (1, 0, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 27. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}\), \(\displaystyle g_{-1}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-5}\), \(\displaystyle g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{3}+\omega_{5}}\oplus V_{\omega_{2}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{4}} \oplus V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+12\psi}\oplus V_{\omega_{3}+\omega_{5}+6\psi}\oplus V_{2\omega_{5}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{5}-6\psi} \oplus V_{\omega_{4}-12\psi}\) Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_5\) ↪ \(E^{1}_6\) 115 out of 119
Subalgebra type: \(\displaystyle A^{1}_5\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_4\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, 0, 0, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_5+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_5\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, 0, 0, 0, -1): 2 Dimension of subalgebra generated by predefined or computed generators: 35. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}\), \(\displaystyle g_{6}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-5}\), \(\displaystyle g_{-6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{5}}\oplus 2V_{\omega_{3}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{\omega_{1}+\omega_{5}}\oplus V_{0}\oplus V_{\omega_{3}-2\psi}\oplus V_{-4\psi}\) Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(D^{1}_5\) ↪ \(E^{1}_6\) 116 out of 119
Subalgebra type: \(\displaystyle D^{1}_5\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_4\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\) Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle D^{1}_5\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, -1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 45. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}\), \(\displaystyle g_{3}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-5}\), \(\displaystyle g_{-3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & -1\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & -1 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & -1\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & -1 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{5}}\oplus V_{\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{5}+6\psi}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{4}-6\psi}\) Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{1}_2\) ↪ \(E^{1}_6\) 117 out of 119
Subalgebra type: \(\displaystyle 3A^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle 2A^{1}_2+A^{1}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \(\displaystyle A^{1}_2\): (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, \(\displaystyle A^{1}_2\): (1, 0, 1, 0, 0, 0): 2, (0, 0, -1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 24. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-11}\), \(\displaystyle g_{6}\), \(\displaystyle g_{-7}\), \(\displaystyle g_{3}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{11}\), \(\displaystyle g_{-6}\), \(\displaystyle g_{7}\), \(\displaystyle g_{-3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 0 & 2 & -1\\
0 & 0 & 0 & 0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 0 & 2 & -1\\
0 & 0 & 0 & 0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{3}+\omega_{6}}\oplus V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{\omega_{5}+\omega_{6}}\oplus V_{\omega_{3}+\omega_{4}} \oplus V_{\omega_{1}+\omega_{2}}\) Made total 701 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_5+A^{1}_1\) ↪ \(E^{1}_6\) 118 out of 119
Subalgebra type: \(\displaystyle A^{1}_5+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_5\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_5\): (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, 0, 0, 0, -1): 2, \(\displaystyle A^{1}_1\): (1, 0, 0, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 38. Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}\), \(\displaystyle g_{6}\), \(\displaystyle g_{-1}\) Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-5}\), \(\displaystyle g_{-6}\), \(\displaystyle g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{6}}\oplus V_{\omega_{3}+\omega_{6}}\oplus V_{\omega_{1}+\omega_{5}}\) Made total 697 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(E^{1}_6\) ↪ \(E^{1}_6\) 119 out of 119
Subalgebra type: \(\displaystyle E^{1}_6\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle D^{1}_5\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)