Subalgebra \(C^{1}_3+A^{10}_1\) ↪ \(C^{1}_5\)
109 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle C^{1}_3+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3\) .
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: \(\displaystyle C^{1}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2, \(\displaystyle A^{10}_1\): (0, 0, 0, 6, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-19}\), \(\displaystyle 4g_{5}+3g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 2 & 0\\ 0 & 0 & 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{4}}\oplus V_{\omega_{1}+3\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{1}}\)
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 4) ; the vectors are over the primal subalgebra.\(g_{25}\)\(g_{5}+3/4g_{4}\)\(g_{20}\)\(g_{13}\)
weight\(2\omega_{1}\)\(2\omega_{4}\)\(\omega_{1}+3\omega_{4}\)\(6\omega_{4}\)
Isotypic module decomposition over primal subalgebra (total 4 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0, 0, 0)\(\displaystyle V_{2\omega_{4}} \) → (0, 0, 0, 2)\(\displaystyle V_{\omega_{1}+3\omega_{4}} \) → (1, 0, 0, 3)\(\displaystyle V_{6\omega_{4}} \) → (0, 0, 0, 6)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{25}\)
\(g_{1}\)
\(2g_{-23}\)
\(-g_{6}\)
\(-2g_{-21}\)
\(g_{22}\)
\(4g_{-19}\)
\(-2g_{-2}\)
\(g_{24}\)
\(4h_{5}+8h_{4}+8h_{3}\)
\(-4h_{2}\)
\(-2h_{5}-4h_{4}-4h_{3}-4h_{2}-2h_{1}\)
\(4g_{2}\)
\(-8g_{19}\)
\(-2g_{-24}\)
\(4g_{-22}\)
\(-4g_{21}\)
\(4g_{-6}\)
\(-8g_{23}\)
\(4g_{-1}\)
\(8g_{-25}\)
Semisimple subalgebra component.
\(-4/3g_{5}-g_{4}\)
\(4/3h_{5}+2h_{4}\)
\(2/3g_{-4}+2/3g_{-5}\)
\(g_{20}\)
\(-g_{-7}\)
\(g_{17}\)
\(g_{-3}\)
\(-g_{-11}\)
\(-g_{14}\)
\(-g_{16}\)
\(g_{-8}\)
\(-g_{-15}\)
\(g_{10}\)
\(-g_{18}\)
\(-g_{12}\)
\(g_{-12}\)
\(g_{-18}\)
\(g_{-10}\)
\(-g_{15}\)
\(g_{8}\)
\(-g_{-16}\)
\(g_{-14}\)
\(g_{11}\)
\(-g_{3}\)
\(g_{-17}\)
\(-g_{7}\)
\(-g_{-20}\)
\(g_{13}\)
\(g_{9}\)
\(2g_{5}-g_{4}\)
\(-2h_{5}+2h_{4}\)
\(4g_{-4}-6g_{-5}\)
\(10g_{-9}\)
\(-20g_{-13}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(2\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(2\omega_{1}-2\omega_{2}\)
\(-\omega_{2}\)
\(-2\omega_{1}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)		\(\omega_{1}+3\omega_{4}\)
\(-\omega_{1}+\omega_{2}+3\omega_{4}\)
\(\omega_{1}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}+3\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(\omega_{2}-\omega_{3}+3\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(\omega_{1}-3\omega_{4}\)
\(\omega_{1}-\omega_{2}+3\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-3\omega_{4}\)
\(-\omega_{1}+3\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}-3\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(\omega_{2}-\omega_{3}-3\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)
\(\omega_{1}-\omega_{2}-3\omega_{4}\)
\(-\omega_{1}-3\omega_{4}\)		\(6\omega_{4}\)
\(4\omega_{4}\)
\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)
\(-4\omega_{4}\)
\(-6\omega_{4}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(2\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(2\omega_{1}-2\omega_{2}\)
\(-\omega_{2}\)
\(-2\omega_{1}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)		\(\omega_{1}+3\omega_{4}\)
\(-\omega_{1}+\omega_{2}+3\omega_{4}\)
\(\omega_{1}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}+3\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(\omega_{2}-\omega_{3}+3\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(\omega_{1}-3\omega_{4}\)
\(\omega_{1}-\omega_{2}+3\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-3\omega_{4}\)
\(-\omega_{1}+3\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}-3\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(\omega_{2}-\omega_{3}-3\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)
\(\omega_{1}-\omega_{2}-3\omega_{4}\)
\(-\omega_{1}-3\omega_{4}\)		\(6\omega_{4}\)
\(4\omega_{4}\)
\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)
\(-4\omega_{4}\)
\(-6\omega_{4}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+2\omega_{2}}
\oplus 3M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{2\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}}\)		\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)\(\displaystyle M_{\omega_{1}+3\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+3\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+3\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+3\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+3\omega_{4}}\oplus M_{-\omega_{1}+3\omega_{4}}
\oplus M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}-\omega_{4}}\oplus M_{\omega_{1}-3\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-3\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-3\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}-3\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-3\omega_{4}}\oplus M_{-\omega_{1}-3\omega_{4}}\)		\(\displaystyle M_{6\omega_{4}}\oplus M_{4\omega_{4}}\oplus M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\oplus M_{-4\omega_{4}}\oplus M_{-6\omega_{4}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+2\omega_{2}}
\oplus 3M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{2\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}}\)		\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)\(\displaystyle M_{\omega_{1}+3\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+3\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+3\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+3\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+3\omega_{4}}\oplus M_{-\omega_{1}+3\omega_{4}}
\oplus M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}-\omega_{4}}\oplus M_{\omega_{1}-3\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-3\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-3\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}-3\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-3\omega_{4}}\oplus M_{-\omega_{1}-3\omega_{4}}\)		\(\displaystyle M_{6\omega_{4}}\oplus M_{4\omega_{4}}\oplus M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\oplus M_{-4\omega_{4}}\oplus M_{-6\omega_{4}}\)

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 3334 arithmetic operations while solving the Serre relations polynomial system.