Subalgebra \(A^{35}_1+A^{1}_1\) ↪ \(E^{1}_6\)
44 out of 119
Computations done by the calculator project.

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: \(\displaystyle A^{35}_1\): (8, 10, 14, 19, 14, 8): 70, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-6}+g_{-13}+g_{-14}+g_{-15}\), \(\displaystyle g_{-4}\)
Positive simple generators: \(\displaystyle 9g_{15}+5g_{14}+5g_{13}+8g_{6}+8g_{1}\), \(\displaystyle g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{9\omega_{1}+\omega_{2}}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}
\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
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 9) ; the vectors are over the primal subalgebra.\(g_{15}+5/9g_{14}+5/9g_{13}+8/9g_{6}+8/9g_{1}\)\(g_{4}\)\(-g_{21}+5/9g_{20}-g_{18}+5/9g_{17}\)\(-g_{23}-8/5g_{16}+8/5g_{12}\)\(-g_{31}-g_{29}+8/5g_{24}\)\(g_{28}+g_{26}\)\(g_{33}+g_{32}\)\(g_{36}\)\(g_{35}\)
weight\(2\omega_{1}\)\(2\omega_{2}\)\(4\omega_{1}\)\(3\omega_{1}+\omega_{2}\)\(6\omega_{1}\)\(5\omega_{1}+\omega_{2}\)\(8\omega_{1}\)\(10\omega_{1}\)\(9\omega_{1}+\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 9 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0)\(\displaystyle V_{2\omega_{2}} \) → (0, 2)\(\displaystyle V_{4\omega_{1}} \) → (4, 0)\(\displaystyle V_{3\omega_{1}+\omega_{2}} \) → (3, 1)\(\displaystyle V_{6\omega_{1}} \) → (6, 0)\(\displaystyle V_{5\omega_{1}+\omega_{2}} \) → (5, 1)\(\displaystyle V_{8\omega_{1}} \) → (8, 0)\(\displaystyle V_{10\omega_{1}} \) → (10, 0)\(\displaystyle V_{9\omega_{1}+\omega_{2}} \) → (9, 1)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)\(W_{8}\)\(W_{9}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-9/8g_{15}-5/8g_{14}-5/8g_{13}-g_{6}-g_{1}\)
\(h_{6}+7/4h_{5}+19/8h_{4}+7/4h_{3}+5/4h_{2}+h_{1}\)
\(1/4g_{-1}+1/4g_{-6}+1/4g_{-13}+1/4g_{-14}+1/4g_{-15}\)
Semisimple subalgebra component.
\(-g_{4}\)
\(h_{4}\)
\(2g_{-4}\)
\(-g_{21}+5/9g_{20}-g_{18}+5/9g_{17}\)
\(-5/9g_{14}+5/9g_{13}-4/9g_{6}+4/9g_{1}\)
\(4/9h_{6}+5/9h_{5}-5/9h_{3}-4/9h_{1}\)
\(-1/3g_{-1}+1/3g_{-6}-2/3g_{-13}+2/3g_{-14}\)
\(1/3g_{-17}-1/3g_{-18}+1/3g_{-20}-1/3g_{-21}\)
\(-g_{23}-8/5g_{16}+8/5g_{12}\)
\(3/5g_{10}+3/5g_{9}-g_{8}\)
\(-g_{19}-8/5g_{11}-8/5g_{7}\)
\(-6/5g_{-2}+2/5g_{-3}-2/5g_{-5}\)
\(3/5g_{5}-3/5g_{3}+g_{2}\)
\(-2/5g_{-7}-2/5g_{-11}-2/5g_{-19}\)
\(-6/5g_{-8}+2/5g_{-9}+2/5g_{-10}\)
\(-2/5g_{-12}+2/5g_{-16}+2/5g_{-23}\)
\(-g_{31}-g_{29}+8/5g_{24}\)
\(3/5g_{21}-g_{20}-3/5g_{18}+g_{17}\)
\(-6/5g_{15}+g_{14}+g_{13}-2/5g_{6}-2/5g_{1}\)
\(2/5h_{6}+1/5h_{5}-4/5h_{4}+1/5h_{3}-2h_{2}+2/5h_{1}\)
\(3/5g_{-1}+3/5g_{-6}-12/5g_{-13}-12/5g_{-14}+8/5g_{-15}\)
\(3g_{-17}-g_{-18}-3g_{-20}+g_{-21}\)
\(-2g_{-24}+2g_{-29}+2g_{-31}\)
\(g_{28}+g_{26}\)
\(g_{16}+g_{12}\)
\(g_{25}+g_{22}\)
\(-g_{10}+g_{9}\)
\(g_{11}-g_{7}\)
\(g_{-3}+g_{-5}\)
\(-g_{5}-g_{3}\)
\(-g_{-7}+g_{-11}\)
\(g_{-9}-g_{-10}\)
\(-g_{-22}-g_{-25}\)
\(-g_{-12}-g_{-16}\)
\(g_{-26}+g_{-28}\)
\(g_{33}+g_{32}\)
\(g_{31}-g_{29}\)
\(g_{21}+g_{20}+g_{18}+g_{17}\)
\(-g_{14}+g_{13}+2g_{6}-2g_{1}\)
\(-2h_{6}+h_{5}-h_{3}+2h_{1}\)
\(5g_{-1}-5g_{-6}-4g_{-13}+4g_{-14}\)
\(9g_{-17}+5g_{-18}+9g_{-20}+5g_{-21}\)
\(14g_{-29}-14g_{-31}\)
\(14g_{-32}+14g_{-33}\)
\(g_{36}\)
\(-g_{33}+g_{32}\)
\(-g_{31}-g_{29}-2g_{24}\)
\(-3g_{21}-g_{20}+3g_{18}+g_{17}\)
\(6g_{15}+g_{14}+g_{13}-4g_{6}-4g_{1}\)
\(4h_{6}-7h_{5}-8h_{4}-7h_{3}-2h_{2}+4h_{1}\)
\(15g_{-1}+15g_{-6}-6g_{-13}-6g_{-14}-20g_{-15}\)
\(21g_{-17}+35g_{-18}-21g_{-20}-35g_{-21}\)
\(70g_{-24}+56g_{-29}+56g_{-31}\)
\(126g_{-32}-126g_{-33}\)
\(-252g_{-36}\)
\(g_{35}\)
\(-g_{30}\)
\(g_{34}\)
\(-g_{28}+g_{26}\)
\(-g_{27}\)
\(2g_{23}-g_{16}+g_{12}\)
\(-g_{25}+g_{22}\)
\(3g_{10}+3g_{9}+2g_{8}\)
\(2g_{19}-g_{11}-g_{7}\)
\(-6g_{-2}-5g_{-3}+5g_{-5}\)
\(3g_{5}-3g_{3}-2g_{2}\)
\(5g_{-7}+5g_{-11}-16g_{-19}\)
\(-6g_{-8}-5g_{-9}-5g_{-10}\)
\(21g_{-22}-21g_{-25}\)
\(5g_{-12}-5g_{-16}+16g_{-23}\)
\(42g_{-27}\)
\(-21g_{-26}+21g_{-28}\)
\(42g_{-34}\)
\(-42g_{-30}\)
\(-42g_{-35}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(5\omega_{1}+\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(9\omega_{1}+\omega_{2}\)
\(7\omega_{1}+\omega_{2}\)
\(9\omega_{1}-\omega_{2}\)
\(5\omega_{1}+\omega_{2}\)
\(7\omega_{1}-\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-7\omega_{1}+\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)
\(-9\omega_{1}+\omega_{2}\)
\(-7\omega_{1}-\omega_{2}\)
\(-9\omega_{1}-\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(5\omega_{1}+\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(9\omega_{1}+\omega_{2}\)
\(7\omega_{1}+\omega_{2}\)
\(9\omega_{1}-\omega_{2}\)
\(5\omega_{1}+\omega_{2}\)
\(7\omega_{1}-\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-7\omega_{1}+\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)
\(-9\omega_{1}+\omega_{2}\)
\(-7\omega_{1}-\omega_{2}\)
\(-9\omega_{1}-\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{5\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{9\omega_{1}+\omega_{2}}\oplus M_{7\omega_{1}+\omega_{2}}\oplus M_{9\omega_{1}-\omega_{2}}\oplus M_{5\omega_{1}+\omega_{2}}
\oplus M_{7\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-7\omega_{1}+\omega_{2}}
\oplus M_{-5\omega_{1}-\omega_{2}}\oplus M_{-9\omega_{1}+\omega_{2}}\oplus M_{-7\omega_{1}-\omega_{2}}\oplus M_{-9\omega_{1}-\omega_{2}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{5\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{9\omega_{1}+\omega_{2}}\oplus M_{7\omega_{1}+\omega_{2}}\oplus M_{9\omega_{1}-\omega_{2}}\oplus M_{5\omega_{1}+\omega_{2}}
\oplus M_{7\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-7\omega_{1}+\omega_{2}}
\oplus M_{-5\omega_{1}-\omega_{2}}\oplus M_{-9\omega_{1}+\omega_{2}}\oplus M_{-7\omega_{1}-\omega_{2}}\oplus M_{-9\omega_{1}-\omega_{2}}\)

Semisimple subalgebra: W_{1}+W_{2}
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 727 arithmetic operations while solving the Serre relations polynomial system.