Subalgebra \(A^{8}_1+A^{3}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
98 out of 119
Computations done by the calculator project.

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

Semisimple subalgebra: W_{1}+W_{2}+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 723 arithmetic operations while solving the Serre relations polynomial system.