Subalgebra \(G^{1}_2\) ↪ \(B^{1}_3\)
13 out of 16
Computations done by the calculator project.

Subalgebra type: \(\displaystyle G^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_3\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (2, 3, 4): 6, (-1, -1, -2): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \(\displaystyle g_{-6}+g_{-7}\), \(\displaystyle g_{8}\)
Positive simple generators: \(\displaystyle g_{7}+g_{6}\), \(\displaystyle g_{-8}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1\\ -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\ -3 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}}\oplus V_{\omega_{1}}\)
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 2) ; the vectors are over the primal subalgebra.\(g_{5}+2g_{4}\)\(g_{2}\)
weight\(\omega_{1}\)\(\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 2 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}} \) → (1, 0)\(\displaystyle V_{\omega_{2}} \) → (0, 1)
Module label \(W_{1}\)\(W_{2}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
\(g_{5}+2g_{4}\)
\(-2g_{-1}+g_{-3}\)
\(2g_{7}-g_{6}\)
\(-2h_{3}+2h_{1}\)
\(2g_{-6}-4g_{-7}\)
\(2g_{3}-4g_{1}\)
\(-4g_{-4}-2g_{-5}\)
Semisimple subalgebra component.
\(-g_{2}\)
\(g_{9}\)
\(-g_{5}+g_{4}\)
\(2g_{-1}+2g_{-3}\)
\(6g_{-8}\)
\(-2g_{7}-2g_{6}\)
\(12h_{3}+6h_{2}+6h_{1}\)
\(8h_{3}+6h_{2}+4h_{1}\)
\(6g_{-6}+6g_{-7}\)
\(-12g_{8}\)
\(6g_{3}+6g_{1}\)
\(-12g_{-4}+12g_{-5}\)
\(36g_{-9}\)
\(36g_{-2}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)		\(\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-3\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(3\omega_{1}-2\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)		\(\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-3\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(3\omega_{1}-2\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\)		\(\displaystyle M_{3\omega_{1}-\omega_{2}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus 2M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\)
Isotypic character\(\displaystyle M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\)		\(\displaystyle M_{3\omega_{1}-\omega_{2}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus 2M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\)

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 450 arithmetic operations while solving the Serre relations polynomial system.