Subalgebra \(A^{1}_2+A^{8}_1\) ↪ \(E^{1}_6\)
71 out of 119
Computations done by the calculator project.

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: \(\displaystyle V_{\omega_{2}+4\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus 2V_{4\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}}
\oplus 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\)
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.
Highest vectors of representations (total 11) ; the vectors are over the primal subalgebra.\(-g_{31}+g_{30}+g_{29}\)\(g_{16}+g_{15}+g_{12}\)\(g_{35}\)\(g_{3}+g_{1}\)\(g_{6}+g_{5}\)\(-g_{33}+g_{32}\)\(g_{21}+g_{18}\)\(g_{11}\)\(g_{7}\)\(g_{34}\)\(g_{24}\)
weight\(\omega_{1}\)\(\omega_{2}\)\(\omega_{1}+\omega_{2}\)\(2\omega_{3}\)\(2\omega_{3}\)\(\omega_{1}+2\omega_{3}\)\(\omega_{2}+2\omega_{3}\)\(4\omega_{3}\)\(4\omega_{3}\)\(\omega_{1}+4\omega_{3}\)\(\omega_{2}+4\omega_{3}\)
Isotypic module decomposition over primal subalgebra (total 10 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}} \) → (1, 0, 0)\(\displaystyle V_{\omega_{2}} \) → (0, 1, 0)\(\displaystyle V_{\omega_{1}+\omega_{2}} \) → (1, 1, 0)\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2)\(\displaystyle V_{\omega_{1}+2\omega_{3}} \) → (1, 0, 2)\(\displaystyle V_{\omega_{2}+2\omega_{3}} \) → (0, 1, 2)\(\displaystyle V_{4\omega_{3}} \) → (0, 0, 4)\(\displaystyle V_{\omega_{1}+4\omega_{3}} \) → (1, 0, 4)\(\displaystyle V_{\omega_{2}+4\omega_{3}} \) → (0, 1, 4)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)\(W_{8}\)\(W_{9}\)\(W_{10}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
\(-g_{31}+g_{30}+g_{29}\)
\(g_{-17}+g_{-19}+g_{-20}\)
\(-g_{-12}-g_{-15}-g_{-16}\)
\(g_{16}+g_{15}+g_{12}\)
\(g_{20}+g_{19}+g_{17}\)
\(-g_{-29}-g_{-30}+g_{-31}\)
Semisimple subalgebra component.
\(-g_{35}\)
\(g_{-2}\)
\(-g_{36}\)
\(h_{2}\)
\(h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}\)
\(g_{-36}\)
\(-2g_{2}\)
\(-g_{-35}\)
Semisimple subalgebra component.
\(g_{6}+g_{5}+g_{3}+g_{1}\)
\(-h_{6}-h_{5}-h_{3}-h_{1}\)
\(-g_{-1}-g_{-3}-g_{-5}-g_{-6}\)
\(g_{3}+g_{1}\)
\(-h_{3}-h_{1}\)
\(-g_{-1}-g_{-3}\)
\(-g_{33}+g_{32}\)
\(-g_{-13}-g_{-14}\)
\(-g_{31}-g_{29}\)
\(g_{-9}+g_{-10}\)
\(g_{-17}-g_{-20}\)
\(-g_{28}-g_{26}\)
\(-g_{-12}+g_{-16}\)
\(g_{-22}+g_{-25}\)
\(-g_{-18}-g_{-21}\)
\(g_{21}+g_{18}\)
\(g_{25}+g_{22}\)
\(g_{16}-g_{12}\)
\(g_{-26}+g_{-28}\)
\(g_{20}-g_{17}\)
\(-g_{10}-g_{9}\)
\(-g_{-29}-g_{-31}\)
\(-g_{14}-g_{13}\)
\(-g_{-32}+g_{-33}\)
\(g_{11}\)
\(g_{6}-g_{5}\)
\(-h_{6}+h_{5}\)
\(3g_{-5}-3g_{-6}\)
\(6g_{-11}\)
\(g_{7}\)
\(g_{3}-g_{1}\)
\(-h_{3}+h_{1}\)
\(3g_{-1}-3g_{-3}\)
\(6g_{-7}\)
\(g_{34}\)
\(-g_{-8}\)
\(g_{33}+g_{32}\)
\(g_{-4}\)
\(g_{-13}-g_{-14}\)
\(g_{31}+2g_{30}-g_{29}\)
\(-g_{-9}+g_{-10}\)
\(-g_{-17}+2g_{-19}-g_{-20}\)
\(3g_{28}-3g_{26}\)
\(g_{-12}-2g_{-15}+g_{-16}\)
\(-3g_{-22}+3g_{-25}\)
\(-6g_{23}\)
\(3g_{-18}-3g_{-21}\)
\(-6g_{-27}\)
\(6g_{-24}\)
\(g_{24}\)
\(g_{27}\)
\(g_{21}-g_{18}\)
\(-g_{-23}\)
\(g_{25}-g_{22}\)
\(g_{16}-2g_{15}+g_{12}\)
\(g_{-26}-g_{-28}\)
\(g_{20}-2g_{19}+g_{17}\)
\(-3g_{10}+3g_{9}\)
\(-g_{-29}+2g_{-30}+g_{-31}\)
\(-3g_{14}+3g_{13}\)
\(6g_{4}\)
\(-3g_{-32}-3g_{-33}\)
\(6g_{8}\)
\(6g_{-34}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)		\(\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{2}-2\omega_{3}\)		\(\omega_{2}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(\omega_{2}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(\omega_{2}-2\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}-2\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)		\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)		\(\omega_{1}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+4\omega_{3}\)
\(\omega_{1}+2\omega_{3}\)
\(-\omega_{2}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(\omega_{1}-4\omega_{3}\)
\(-\omega_{2}-2\omega_{3}\)
\(-\omega_{1}+\omega_{2}-4\omega_{3}\)
\(-\omega_{2}-4\omega_{3}\)		\(\omega_{2}+4\omega_{3}\)
\(\omega_{1}-\omega_{2}+4\omega_{3}\)
\(\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+4\omega_{3}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(\omega_{2}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(\omega_{2}-2\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}-2\omega_{3}\)
\(\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-4\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)		\(\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{2}-2\omega_{3}\)		\(\omega_{2}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(\omega_{2}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(\omega_{2}-2\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}-2\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)		\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)		\(\omega_{1}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+4\omega_{3}\)
\(\omega_{1}+2\omega_{3}\)
\(-\omega_{2}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(\omega_{1}-4\omega_{3}\)
\(-\omega_{2}-2\omega_{3}\)
\(-\omega_{1}+\omega_{2}-4\omega_{3}\)
\(-\omega_{2}-4\omega_{3}\)		\(\omega_{2}+4\omega_{3}\)
\(\omega_{1}-\omega_{2}+4\omega_{3}\)
\(\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+4\omega_{3}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(\omega_{2}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(\omega_{2}-2\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}-2\omega_{3}\)
\(\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-4\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\)\(\displaystyle M_{\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}\)\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{1}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{2}-2\omega_{3}}\)		\(\displaystyle M_{\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}\oplus M_{\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-2\omega_{3}}\)		\(\displaystyle M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}\)\(\displaystyle M_{\omega_{1}+4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{2}+4\omega_{3}}\oplus M_{\omega_{1}+2\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{2}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{2}-2\omega_{3}}
\oplus M_{\omega_{1}-4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{2}-4\omega_{3}}\)		\(\displaystyle M_{\omega_{2}+4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{1}+4\omega_{3}}\oplus M_{\omega_{2}+2\omega_{3}}
\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}-2\omega_{3}}
\oplus M_{\omega_{2}-4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{1}-4\omega_{3}}\)
Isotypic character\(\displaystyle M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\)\(\displaystyle M_{\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}\)\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{1}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{2}-2\omega_{3}}\)		\(\displaystyle M_{\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}\oplus M_{\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-2\omega_{3}}\)		\(\displaystyle 2M_{4\omega_{3}}\oplus 2M_{2\omega_{3}}\oplus 2M_{0}\oplus 2M_{-2\omega_{3}}\oplus 2M_{-4\omega_{3}}\)\(\displaystyle M_{\omega_{1}+4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{2}+4\omega_{3}}\oplus M_{\omega_{1}+2\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{2}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{2}-2\omega_{3}}
\oplus M_{\omega_{1}-4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{2}-4\omega_{3}}\)		\(\displaystyle M_{\omega_{2}+4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{1}+4\omega_{3}}\oplus M_{\omega_{2}+2\omega_{3}}
\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}-2\omega_{3}}
\oplus M_{\omega_{2}-4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{1}-4\omega_{3}}\)

Semisimple subalgebra: W_{3}+W_{4}
Centralizer extension: 0

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.
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Made total 30250 arithmetic operations while solving the Serre relations polynomial system.