Subalgebra \(B^{1}_2+A^{35}_1\) ↪ \(C^{1}_5\)
87 out of 119
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Subalgebra type: \(\displaystyle B^{1}_2+A^{35}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
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: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{35}_1\): (0, 0, 10, 16, 9): 70
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-3}+g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 9g_{5}+8g_{4}+5g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2/35\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 70\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{3}}\oplus V_{6\omega_{3}}\oplus V_{\omega_{2}+5\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\)
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 5) ; the vectors are over the primal subalgebra.\(g_{23}\)\(g_{5}+8/9g_{4}+5/9g_{3}\)\(g_{21}\)\(-g_{13}+5/8g_{12}\)\(g_{19}\)
weight\(2\omega_{2}\)\(2\omega_{3}\)\(\omega_{2}+5\omega_{3}\)\(6\omega_{3}\)\(10\omega_{3}\)
Isotypic module decomposition over primal subalgebra (total 5 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{2}} \) → (0, 2, 0)\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2)\(\displaystyle V_{\omega_{2}+5\omega_{3}} \) → (0, 1, 5)\(\displaystyle V_{6\omega_{3}} \) → (0, 0, 6)\(\displaystyle V_{10\omega_{3}} \) → (0, 0, 10)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{23}\)
\(-g_{24}\)
\(g_{-1}\)
\(-2g_{25}\)
\(2h_{1}\)
\(2h_{5}+4h_{4}+4h_{3}+4h_{2}+4h_{1}\)
\(2g_{-25}\)
\(-2g_{1}\)
\(-2g_{-24}\)
\(4g_{-23}\)
Semisimple subalgebra component.
\(-9/5g_{5}-8/5g_{4}-g_{3}\)
\(9/5h_{5}+16/5h_{4}+2h_{3}\)
\(2/5g_{-3}+2/5g_{-4}+2/5g_{-5}\)
\(g_{21}\)
\(g_{22}\)
\(g_{18}\)
\(-g_{-6}\)
\(g_{20}\)
\(g_{15}\)
\(g_{-2}\)
\(-g_{-10}\)
\(g_{17}\)
\(-g_{11}\)
\(g_{-7}\)
\(-g_{-14}\)
\(-g_{14}\)
\(g_{7}\)
\(g_{-11}\)
\(-g_{-17}\)
\(g_{10}\)
\(-g_{2}\)
\(g_{-15}\)
\(g_{-20}\)
\(-g_{6}\)
\(-g_{-18}\)
\(-g_{-22}\)
\(g_{-21}\)
\(-g_{13}+5/8g_{12}\)
\(-3/8g_{9}-5/8g_{8}\)
\(-3/4g_{5}-1/4g_{4}+5/8g_{3}\)
\(3/4h_{5}+1/2h_{4}-5/4h_{3}\)
\(-3/2g_{-3}+3/8g_{-4}+g_{-5}\)
\(-15/8g_{-8}-5/8g_{-9}\)
\(-5/4g_{-12}+5/4g_{-13}\)
\(g_{19}\)
\(g_{16}\)
\(2g_{13}+g_{12}\)
\(3g_{9}-g_{8}\)
\(6g_{5}-4g_{4}+g_{3}\)
\(-6h_{5}+8h_{4}-2h_{3}\)
\(-6g_{-3}+15g_{-4}-20g_{-5}\)
\(-21g_{-8}+35g_{-9}\)
\(-56g_{-12}-70g_{-13}\)
\(126g_{-16}\)
\(-252g_{-19}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-2\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}\)
\(-2\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{2}+5\omega_{3}\)
\(\omega_{1}-\omega_{2}+5\omega_{3}\)
\(\omega_{2}+3\omega_{3}\)
\(-\omega_{1}+\omega_{2}+5\omega_{3}\)
\(\omega_{1}-\omega_{2}+3\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(-\omega_{2}+5\omega_{3}\)
\(-\omega_{1}+\omega_{2}+3\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{2}+3\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(\omega_{2}-3\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{1}-\omega_{2}-3\omega_{3}\)
\(\omega_{2}-5\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-3\omega_{3}\)
\(\omega_{1}-\omega_{2}-5\omega_{3}\)
\(-\omega_{2}-3\omega_{3}\)
\(-\omega_{1}+\omega_{2}-5\omega_{3}\)
\(-\omega_{2}-5\omega_{3}\)		\(6\omega_{3}\)
\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)
\(-6\omega_{3}\)		\(10\omega_{3}\)
\(8\omega_{3}\)
\(6\omega_{3}\)
\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)
\(-6\omega_{3}\)
\(-8\omega_{3}\)
\(-10\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-2\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}\)
\(-2\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{2}+5\omega_{3}\)
\(\omega_{1}-\omega_{2}+5\omega_{3}\)
\(\omega_{2}+3\omega_{3}\)
\(-\omega_{1}+\omega_{2}+5\omega_{3}\)
\(\omega_{1}-\omega_{2}+3\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(-\omega_{2}+5\omega_{3}\)
\(-\omega_{1}+\omega_{2}+3\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{2}+3\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(\omega_{2}-3\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{1}-\omega_{2}-3\omega_{3}\)
\(\omega_{2}-5\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-3\omega_{3}\)
\(\omega_{1}-\omega_{2}-5\omega_{3}\)
\(-\omega_{2}-3\omega_{3}\)
\(-\omega_{1}+\omega_{2}-5\omega_{3}\)
\(-\omega_{2}-5\omega_{3}\)		\(6\omega_{3}\)
\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)
\(-6\omega_{3}\)		\(10\omega_{3}\)
\(8\omega_{3}\)
\(6\omega_{3}\)
\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)
\(-6\omega_{3}\)
\(-8\omega_{3}\)
\(-10\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus 2M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{2}+5\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+5\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+5\omega_{3}}
\oplus M_{-\omega_{2}+5\omega_{3}}\oplus M_{\omega_{2}+3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+3\omega_{3}}
\oplus M_{-\omega_{2}+3\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-3\omega_{3}}
\oplus M_{-\omega_{2}-3\omega_{3}}\oplus M_{\omega_{2}-5\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-5\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-5\omega_{3}}
\oplus M_{-\omega_{2}-5\omega_{3}}\)		\(\displaystyle M_{6\omega_{3}}\oplus M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}\oplus M_{-6\omega_{3}}\)\(\displaystyle M_{10\omega_{3}}\oplus M_{8\omega_{3}}\oplus M_{6\omega_{3}}\oplus M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}
\oplus M_{-6\omega_{3}}\oplus M_{-8\omega_{3}}\oplus M_{-10\omega_{3}}\)
Isotypic character\(\displaystyle M_{2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus 2M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{2}+5\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+5\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+5\omega_{3}}
\oplus M_{-\omega_{2}+5\omega_{3}}\oplus M_{\omega_{2}+3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+3\omega_{3}}
\oplus M_{-\omega_{2}+3\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-3\omega_{3}}
\oplus M_{-\omega_{2}-3\omega_{3}}\oplus M_{\omega_{2}-5\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-5\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-5\omega_{3}}
\oplus M_{-\omega_{2}-5\omega_{3}}\)		\(\displaystyle M_{6\omega_{3}}\oplus M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}\oplus M_{-6\omega_{3}}\)\(\displaystyle M_{10\omega_{3}}\oplus M_{8\omega_{3}}\oplus M_{6\omega_{3}}\oplus M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}
\oplus M_{-6\omega_{3}}\oplus M_{-8\omega_{3}}\oplus M_{-10\omega_{3}}\)

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 10737 arithmetic operations while solving the Serre relations polynomial system.