Subalgebra \(A^{11}_1+A^{10}_1\) ↪ \(C^{1}_5\)
48 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{11}_1+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1\) .
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: \(\displaystyle A^{11}_1\): (6, 8, 10, 10, 5): 22, \(\displaystyle A^{10}_1\): (0, 0, 0, 6, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-19}+g_{-23}\), \(\displaystyle g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{19}+3g_{1}\), \(\displaystyle 4g_{5}+3g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0\\ 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0\\ 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{2}}\oplus V_{3\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}
\oplus 3V_{2\omega_{1}}\)
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.\(g_{21}+3g_{6}\)\(g_{23}+3/4g_{1}\)\(g_{19}\)\(g_{5}+3/4g_{4}\)\(g_{22}\)\(g_{16}\)\(g_{25}\)\(g_{20}\)\(g_{13}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{2}\)\(4\omega_{1}\)\(\omega_{1}+3\omega_{2}\)\(6\omega_{1}\)\(3\omega_{1}+3\omega_{2}\)\(6\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 8 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)\(\displaystyle V_{\omega_{1}+3\omega_{2}} \) → (1, 3)\(\displaystyle V_{6\omega_{1}} \) → (6, 0)\(\displaystyle V_{3\omega_{1}+3\omega_{2}} \) → (3, 3)\(\displaystyle V_{6\omega_{2}} \) → (0, 6)
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. Semisimple subalgebra component.
\(4g_{23}+g_{19}+3g_{1}\)
\(-5h_{5}-10h_{4}-10h_{3}-8h_{2}-6h_{1}\)
\(-2g_{-1}-2g_{-19}-2g_{-23}\)
\(g_{23}+3/4g_{1}\)
\(-h_{5}-2h_{4}-2h_{3}-2h_{2}-3/2h_{1}\)
\(-1/2g_{-1}-1/2g_{-23}\)
\(g_{21}+3g_{6}\)
\(2g_{2}-g_{-2}\)
\(g_{-6}+g_{-21}\)
Semisimple subalgebra component.
\(-4/3g_{5}-g_{4}\)
\(4/3h_{5}+2h_{4}\)
\(2/3g_{-4}+2/3g_{-5}\)
\(g_{22}\)
\(g_{21}-g_{6}\)
\(-2g_{2}-g_{-2}\)
\(g_{-6}-3g_{-21}\)
\(4g_{-22}\)
\(g_{16}\)
\(-g_{-3}\)
\(g_{12}\)
\(-g_{-8}\)
\(-g_{8}\)
\(-g_{-12}\)
\(g_{3}\)
\(g_{-16}\)
\(g_{25}\)
\(g_{24}\)
\(2g_{23}-g_{1}\)
\(-2h_{5}-4h_{4}-4h_{3}-4h_{2}+2h_{1}\)
\(4g_{-1}-6g_{-23}\)
\(10g_{-24}\)
\(-20g_{-25}\)
\(g_{20}\)
\(g_{18}\)
\(g_{17}\)
\(-g_{-7}\)
\(g_{15}\)
\(-g_{14}\)
\(g_{-10}\)
\(-g_{-11}\)
\(-g_{11}\)
\(g_{10}\)
\(g_{-14}\)
\(-g_{-15}\)
\(g_{7}\)
\(g_{-17}\)
\(g_{-18}\)
\(-g_{-20}\)
\(g_{13}\)
\(g_{9}\)
\(2g_{5}-g_{4}\)
\(-2h_{5}+2h_{4}\)
\(4g_{-4}-6g_{-5}\)
\(10g_{-9}\)
\(-20g_{-13}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(\omega_{1}+3\omega_{2}\)
\(-\omega_{1}+3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(\omega_{1}-3\omega_{2}\)
\(-\omega_{1}-3\omega_{2}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(3\omega_{1}+3\omega_{2}\)
\(\omega_{1}+3\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(-\omega_{1}+3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+3\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(3\omega_{1}-3\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(\omega_{1}-3\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-\omega_{1}-3\omega_{2}\)
\(-3\omega_{1}-3\omega_{2}\)		\(6\omega_{2}\)
\(4\omega_{2}\)
\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)
\(-4\omega_{2}\)
\(-6\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(\omega_{1}+3\omega_{2}\)
\(-\omega_{1}+3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(\omega_{1}-3\omega_{2}\)
\(-\omega_{1}-3\omega_{2}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(3\omega_{1}+3\omega_{2}\)
\(\omega_{1}+3\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(-\omega_{1}+3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+3\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(3\omega_{1}-3\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(\omega_{1}-3\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-\omega_{1}-3\omega_{2}\)
\(-3\omega_{1}-3\omega_{2}\)		\(6\omega_{2}\)
\(4\omega_{2}\)
\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)
\(-4\omega_{2}\)
\(-6\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{\omega_{1}+3\omega_{2}}\oplus M_{-\omega_{1}+3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}-3\omega_{2}}\oplus M_{-\omega_{1}-3\omega_{2}}\)		\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{3\omega_{1}+3\omega_{2}}\oplus M_{\omega_{1}+3\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+3\omega_{2}}
\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+3\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}-3\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}
\oplus M_{\omega_{1}-3\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-3\omega_{2}}\oplus M_{-3\omega_{1}-3\omega_{2}}\)		\(\displaystyle M_{6\omega_{2}}\oplus M_{4\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{2}}\oplus M_{-6\omega_{2}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{\omega_{1}+3\omega_{2}}\oplus M_{-\omega_{1}+3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}-3\omega_{2}}\oplus M_{-\omega_{1}-3\omega_{2}}\)		\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{3\omega_{1}+3\omega_{2}}\oplus M_{\omega_{1}+3\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+3\omega_{2}}
\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+3\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}-3\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}
\oplus M_{\omega_{1}-3\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-3\omega_{2}}\oplus M_{-3\omega_{1}-3\omega_{2}}\)		\(\displaystyle M_{6\omega_{2}}\oplus M_{4\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{2}}\oplus M_{-6\omega_{2}}\)

Semisimple subalgebra: W_{1}+W_{3}
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 4368 arithmetic operations while solving the Serre relations polynomial system.