Subalgebra \(C^{1}_3\) ↪ \(A^{1}_5\)
32 out of 37
Computations done by the calculator project.

Subalgebra type: \(\displaystyle C^{1}_3\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\)

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

Semisimple subalgebra: 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 620 arithmetic operations while solving the Serre relations polynomial system.