Subalgebra \(A^{3}_2\) ↪ \(E^{1}_6\)
52 out of 119
Computations done by the calculator project.

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

Semisimple subalgebra: 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 141240 arithmetic operations while solving the Serre relations polynomial system.