Subalgebra \(A^{5}_1+2A^{4}_1\) ↪ \(C^{1}_5\)
69 out of 119
Computations done by the calculator project.

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

Semisimple subalgebra: W_{1}+W_{3}+W_{5}
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 9833 arithmetic operations while solving the Serre relations polynomial system.