Subalgebra \(3A^{4}_1+A^{1}_1\) ↪ \(C^{1}_5\)
97 out of 119
Computations done by the calculator project.

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

Semisimple subalgebra: W_{1}+W_{2}+W_{3}+W_{4}
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 802 arithmetic operations while solving the Serre relations polynomial system.