Subalgebra \(A^{8}_1+A^{3}_1+A^{1}_1\) ↪ \(C^{1}_4\)
34 out of 46
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{3}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)

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

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