Subalgebra \(A^{35}_1+A^{10}_1\) ↪ \(C^{1}_5\)
55 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{35}_1+A^{10}_1\) (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: \(\displaystyle C^{1}_5\)

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

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 3340 arithmetic operations while solving the Serre relations polynomial system.