Subalgebra \(3A^{2}_1\) ↪ \(E^{1}_6\)
60 out of 119

Computations done by the calculator project.

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). \(\displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}
\oplus 2V_{2\omega_{3}}\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\)
In the table below we indicate the highest weight vectors of the decomposition of the ambient Lie algebra as a module over the semisimple part. The second row indicates weights of the highest weight vectors relative to the Cartan of the semisimple subalgebra. As the centralizer is well-chosen and the centralizer of our subalgebra is non-trivial, we may in addition split highest weight vectors with the same weight over the semisimple part over the centralizer (recall that the centralizer preserves the weights over the subalgebra and in particular acts on the highest weight vectors). Therefore we have chosen our highest weight vectors to be, in addition, weight vectors over the Cartan of the centralizer of the starting subalgebra. Their weight over the sum of the Cartans of the semisimple subalgebra and its centralizer is indicated in the third row. The weights corresponding to the Cartan of the centralizer are again indicated with the letter \omega. As there is no preferred way of chosing a basis of the Cartan of the centralizer (unlike the starting semisimple Lie algebra: there we have a preferred basis induced by the fundamental weights), our centralizer weights are simply given by the constant by which the k^th basis element of the Cartan of the centralizer acts on the highest weight vector. Here, we use the choice for basis of the Cartan of the centralizer given at the start of the page.

Highest vectors of representations (total 14) ; the vectors are over the primal subalgebra.

\(-h_{6}-1/2h_{5}+1/2h_{3}+h_{1}\)

\(g_{33}+g_{32}\)

\(g_{34}+g_{30}\)

\(-g_{14}+g_{13}\)

\(g_{19}+g_{8}\)

\(g_{9}\)

\(g_{10}\)

\(g_{31}\)

\(g_{28}\)

\(g_{26}\)

\(g_{29}\)

\(g_{36}\)

\(g_{35}\)

\(g_{23}\)

weight

\(0\)

\(2\omega_{1}\)

\(2\omega_{2}\)

\(2\omega_{3}\)

\(\omega_{1}+\omega_{2}+\omega_{3}\)

\(2\omega_{1}+2\omega_{2}\)

\(2\omega_{1}+2\omega_{3}\)

\(2\omega_{2}+2\omega_{3}\)

weights rel. to Cartan of (centralizer+semisimple s.a.).

\(0\)

\(2\omega_{1}\)

\(2\omega_{2}\)

\(2\omega_{3}\)

\(\omega_{1}+\omega_{2}+\omega_{3}-6\psi\)

\(\omega_{1}+\omega_{2}+\omega_{3}+6\psi\)

\(2\omega_{1}+2\omega_{2}\)

\(2\omega_{1}+2\omega_{3}\)

\(2\omega_{2}+2\omega_{3}\)

Isotypic module decomposition over primal subalgebra (total 12 isotypic components).

Isotypical components + highest weight

\(\displaystyle V_{0} \) → (0, 0, 0, 0)

\(\displaystyle V_{2\omega_{1}} \) → (2, 0, 0, 0)

\(\displaystyle V_{2\omega_{2}} \) → (0, 2, 0, 0)

\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2, 0)

\(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi} \) → (1, 1, 1, -6)

\(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi} \) → (1, 1, 1, 6)

\(\displaystyle V_{2\omega_{1}+2\omega_{2}} \) → (2, 2, 0, 0)

\(\displaystyle V_{2\omega_{1}+2\omega_{3}} \) → (2, 0, 2, 0)

\(\displaystyle V_{2\omega_{2}+2\omega_{3}} \) → (0, 2, 2, 0)

Module label

\(W_{1}\)

\(W_{2}\)

\(W_{3}\)

\(W_{4}\)

\(W_{5}\)

\(W_{6}\)

\(W_{7}\)

\(W_{8}\)

\(W_{9}\)

\(W_{10}\)

\(W_{11}\)

\(W_{12}\)

Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.

Cartan of centralizer component.

\(-h_{6}-1/2h_{5}+1/2h_{3}+h_{1}\)

Semisimple subalgebra component.

\(-g_{34}-g_{30}\)

\(2h_{6}+3h_{5}+4h_{4}+3h_{3}+2h_{2}+2h_{1}\)

\(2g_{-30}+2g_{-34}\)

\(g_{33}+g_{32}\)

\(-g_{5}-g_{3}-g_{-3}-g_{-5}\)

\(-2g_{-32}-2g_{-33}\)

Semisimple subalgebra component.

\(-g_{19}-g_{8}\)

\(h_{5}+2h_{4}+h_{3}+2h_{2}\)

\(2g_{-8}+2g_{-19}\)

\(-g_{14}+g_{13}\)

\(-g_{5}-g_{3}+g_{-3}+g_{-5}\)

\(2g_{-13}-2g_{-14}\)

Semisimple subalgebra component.

\(-g_{10}-g_{9}\)

\(h_{5}+2h_{4}+h_{3}\)

\(2g_{-9}+2g_{-10}\)

\(g_{9}\)

\(-h_{4}-h_{3}\)

\(-2g_{-9}\)

\(g_{31}\)

\(g_{-7}\)

\(-g_{16}\)

\(-g_{25}\)

\(g_{-17}\)

\(g_{-18}\)

\(-g_{6}\)

\(g_{-26}\)

\(g_{28}\)

\(-g_{-1}\)

\(-g_{21}\)

\(-g_{20}\)

\(-g_{-22}\)

\(-g_{-12}\)

\(-g_{11}\)

\(g_{-29}\)

\(g_{26}\)

\(g_{-6}\)

\(-g_{18}\)

\(-g_{17}\)

\(-g_{-25}\)

\(-g_{-16}\)

\(g_{7}\)

\(-g_{-31}\)

\(g_{29}\)

\(g_{-11}\)

\(g_{12}\)

\(g_{22}\)

\(-g_{-20}\)

\(-g_{-21}\)

\(-g_{1}\)

\(-g_{-28}\)

\(g_{36}\)

\(g_{19}-g_{8}\)

\(g_{34}-g_{30}\)

\(2g_{-24}\)

\(-h_{5}-h_{3}\)

\(2g_{24}\)

\(2g_{-30}-2g_{-34}\)

\(2g_{-8}-2g_{-19}\)

\(4g_{-36}\)

\(g_{35}\)

\(g_{15}-g_{4}\)

\(-g_{33}+g_{32}\)

\(-2g_{-27}\)

\(g_{5}-g_{3}+g_{-3}-g_{-5}\)

\(2g_{27}\)

\(2g_{-32}-2g_{-33}\)

\(-2g_{-4}+2g_{-15}\)

\(-4g_{-35}\)

\(g_{23}\)

\(-g_{15}-g_{4}\)

\(-g_{14}-g_{13}\)

\(2g_{-2}\)

\(-g_{5}+g_{3}+g_{-3}-g_{-5}\)

\(2g_{2}\)

\(2g_{-13}+2g_{-14}\)

\(2g_{-4}+2g_{-15}\)

\(4g_{-23}\)

Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above

\(0\)

\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)

\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)

\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)

\(\omega_{1}+\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{2}-\omega_{3}\)

\(2\omega_{1}+2\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-2\omega_{1}-2\omega_{2}\)

\(2\omega_{1}+2\omega_{3}\)
\(2\omega_{3}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{3}\)
\(0\)
\(2\omega_{1}-2\omega_{3}\)
\(-2\omega_{1}\)
\(-2\omega_{3}\)
\(-2\omega_{1}-2\omega_{3}\)

\(2\omega_{2}+2\omega_{3}\)
\(2\omega_{3}\)
\(2\omega_{2}\)
\(-2\omega_{2}+2\omega_{3}\)
\(0\)
\(2\omega_{2}-2\omega_{3}\)
\(-2\omega_{2}\)
\(-2\omega_{3}\)
\(-2\omega_{2}-2\omega_{3}\)

Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer

\(0\)

\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)

\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)

\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)

\(\omega_{1}+\omega_{2}+\omega_{3}-6\psi\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-6\psi\)
\(\omega_{1}-\omega_{2}+\omega_{3}-6\psi\)
\(\omega_{1}+\omega_{2}-\omega_{3}-6\psi\)
\(-\omega_{1}-\omega_{2}+\omega_{3}-6\psi\)
\(-\omega_{1}+\omega_{2}-\omega_{3}-6\psi\)
\(\omega_{1}-\omega_{2}-\omega_{3}-6\psi\)
\(-\omega_{1}-\omega_{2}-\omega_{3}-6\psi\)

\(\omega_{1}+\omega_{2}+\omega_{3}+6\psi\)
\(-\omega_{1}+\omega_{2}+\omega_{3}+6\psi\)
\(\omega_{1}-\omega_{2}+\omega_{3}+6\psi\)
\(\omega_{1}+\omega_{2}-\omega_{3}+6\psi\)
\(-\omega_{1}-\omega_{2}+\omega_{3}+6\psi\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+6\psi\)
\(\omega_{1}-\omega_{2}-\omega_{3}+6\psi\)
\(-\omega_{1}-\omega_{2}-\omega_{3}+6\psi\)

\(2\omega_{1}+2\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-2\omega_{1}-2\omega_{2}\)

\(2\omega_{1}+2\omega_{3}\)
\(2\omega_{3}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{3}\)
\(0\)
\(2\omega_{1}-2\omega_{3}\)
\(-2\omega_{1}\)
\(-2\omega_{3}\)
\(-2\omega_{1}-2\omega_{3}\)

\(2\omega_{2}+2\omega_{3}\)
\(2\omega_{3}\)
\(2\omega_{2}\)
\(-2\omega_{2}+2\omega_{3}\)
\(0\)
\(2\omega_{2}-2\omega_{3}\)
\(-2\omega_{2}\)
\(-2\omega_{3}\)
\(-2\omega_{2}-2\omega_{3}\)

Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.

\(\displaystyle M_{0}\)

\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)

\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)

\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)

\(\displaystyle M_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-6\psi}
\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}-6\psi}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}-6\psi}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}-6\psi}
\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}-6\psi}\oplus M_{-\omega_{1}-\omega_{2}-\omega_{3}-6\psi}\)

\(\displaystyle M_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}+6\psi}
\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}+6\psi}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}+6\psi}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+6\psi}
\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+6\psi}\oplus M_{-\omega_{1}-\omega_{2}-\omega_{3}+6\psi}\)

\(\displaystyle M_{2\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{2}}\oplus M_{-2\omega_{1}-2\omega_{2}}\)

\(\displaystyle M_{2\omega_{1}+2\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{2\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{3}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{3}}
\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{3}}\oplus M_{-2\omega_{1}-2\omega_{3}}\)

\(\displaystyle M_{2\omega_{2}+2\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{2\omega_{2}}\oplus M_{-2\omega_{2}+2\omega_{3}}\oplus M_{0}\oplus M_{2\omega_{2}-2\omega_{3}}
\oplus M_{-2\omega_{2}}\oplus M_{-2\omega_{3}}\oplus M_{-2\omega_{2}-2\omega_{3}}\)

Isotypic character

\(\displaystyle M_{0}\)

\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)

\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)

\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)

\(\displaystyle 2M_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\oplus 2M_{-\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\oplus 2M_{\omega_{1}-\omega_{2}+\omega_{3}-6\psi}
\oplus 2M_{\omega_{1}+\omega_{2}-\omega_{3}-6\psi}\oplus 2M_{-\omega_{1}-\omega_{2}+\omega_{3}-6\psi}\oplus 2M_{-\omega_{1}+\omega_{2}-\omega_{3}-6\psi}
\oplus 2M_{\omega_{1}-\omega_{2}-\omega_{3}-6\psi}\oplus 2M_{-\omega_{1}-\omega_{2}-\omega_{3}-6\psi}\)

\(\displaystyle 2M_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus 2M_{-\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus 2M_{\omega_{1}-\omega_{2}+\omega_{3}+6\psi}
\oplus 2M_{\omega_{1}+\omega_{2}-\omega_{3}+6\psi}\oplus 2M_{-\omega_{1}-\omega_{2}+\omega_{3}+6\psi}\oplus 2M_{-\omega_{1}+\omega_{2}-\omega_{3}+6\psi}
\oplus 2M_{\omega_{1}-\omega_{2}-\omega_{3}+6\psi}\oplus 2M_{-\omega_{1}-\omega_{2}-\omega_{3}+6\psi}\)

Semisimple subalgebra: W_{2}+W_{4}+W_{6}
Centralizer extension: W_{1}

Weight diagram. The coordinates corresponding to the simple roots of the subalgerba are fundamental.
The bilinear form is therefore given relative to the fundamental coordinates.

Made total 73384 arithmetic operations while solving the Serre relations polynomial system.