Subalgebra \(B^{1}_2+A^{2}_1\) ↪ \(D^{1}_4\)
19 out of 23
Computations done by the calculator project.

Subalgebra type: \(\displaystyle B^{1}_2+A^{2}_1\) (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: \(\displaystyle D^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 2, 1, 1): 2, (0, -2, -1, -1): 4, \(\displaystyle A^{2}_1\): (0, 0, 1, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-12}\), \(\displaystyle g_{7}+g_{6}\), \(\displaystyle g_{-3}-g_{-4}\)
Positive simple generators: \(\displaystyle g_{12}\), \(\displaystyle g_{-6}+g_{-7}\), \(\displaystyle -g_{4}+g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\)
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 3) ; the vectors are over the primal subalgebra.\(g_{1}\)\(-g_{4}+g_{3}\)\(g_{11}\)
weight\(2\omega_{2}\)\(2\omega_{3}\)\(\omega_{1}+2\omega_{3}\)
Isotypic module decomposition over primal subalgebra (total 3 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{2}} \) → (0, 2, 0)\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2)\(\displaystyle V_{\omega_{1}+2\omega_{3}} \) → (1, 0, 2)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{1}\)
\(g_{9}+g_{8}\)
\(-g_{-6}-g_{-7}\)
\(2g_{12}\)
\(-h_{4}-h_{3}-2h_{2}\)
\(-2h_{4}-2h_{3}-4h_{2}-2h_{1}\)
\(-2g_{-12}\)
\(2g_{7}+2g_{6}\)
\(2g_{-8}+2g_{-9}\)
\(4g_{-1}\)
Semisimple subalgebra component.
\(g_{4}-g_{3}\)
\(h_{4}+h_{3}\)
\(2g_{-3}-2g_{-4}\)
\(g_{11}\)
\(-g_{-2}\)
\(g_{9}-g_{8}\)
\(g_{4}+g_{3}\)
\(-g_{-6}+g_{-7}\)
\(2g_{5}\)
\(-2g_{10}\)
\(h_{4}-h_{3}\)
\(-2g_{-10}\)
\(-2g_{-5}\)
\(-2g_{7}+2g_{6}\)
\(-2g_{-3}-2g_{-4}\)
\(-2g_{-8}+2g_{-9}\)
\(-4g_{2}\)
\(-4g_{-11}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-2\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}\)
\(-2\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(\omega_{1}-2\omega_{2}+2\omega_{3}\)
\(0\)
\(-\omega_{1}+2\omega_{2}-2\omega_{3}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-2\omega_{2}\)
\(-2\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-2\omega_{2}-2\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-2\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}\)
\(-2\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(\omega_{1}-2\omega_{2}+2\omega_{3}\)
\(0\)
\(-\omega_{1}+2\omega_{2}-2\omega_{3}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-2\omega_{2}\)
\(-2\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-2\omega_{2}-2\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus 2M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{-\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}+2\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}
\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{0}\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{-2\omega_{3}}\oplus M_{-\omega_{1}-2\omega_{3}}
\oplus M_{\omega_{1}-2\omega_{2}-2\omega_{3}}\)
Isotypic character\(\displaystyle M_{2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus 2M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{-\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}+2\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}
\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{0}\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{-2\omega_{3}}\oplus M_{-\omega_{1}-2\omega_{3}}
\oplus M_{\omega_{1}-2\omega_{2}-2\omega_{3}}\)

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 817666 arithmetic operations while solving the Serre relations polynomial system.