Subalgebra \(G^{1}_2+A^{8}_1\) ↪ \(F^{1}_4\)
46 out of 59
Computations done by the calculator project.

Subalgebra type: \(\displaystyle G^{1}_2+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle G^{1}_2\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (3, 6, 8, 4): 6, (-1, -3, -4, -2): 2, \(\displaystyle A^{8}_1\): (0, 0, 4, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 17.
Negative simple generators: \(\displaystyle g_{-19}+g_{-20}\), \(\displaystyle g_{23}\), \(\displaystyle g_{-3}-g_{-4}\)
Positive simple generators: \(\displaystyle g_{20}+g_{19}\), \(\displaystyle g_{-23}\), \(\displaystyle -2g_{4}+2g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1 & 0\\ -1 & 2 & 0\\ 0 & 0 & 1/4\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3 & 0\\ -3 & 2 & 0\\ 0 & 0 & 16\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+4\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}}\)
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 3) ; the vectors are over the primal subalgebra.\(g_{1}\)\(-g_{4}+g_{3}\)\(g_{18}\)
weight\(\omega_{2}\)\(2\omega_{3}\)\(\omega_{1}+4\omega_{3}\)
Isotypic module decomposition over primal subalgebra (total 3 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{2}} \) → (0, 1, 0)\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2)\(\displaystyle V_{\omega_{1}+4\omega_{3}} \) → (1, 0, 4)
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.
\(-g_{1}\)
\(g_{24}\)
\(-g_{12}+g_{11}\)
\(2g_{-9}-2g_{-10}\)
\(-6g_{-23}\)
\(2g_{20}+2g_{19}\)
\(-12h_{4}-24h_{3}-18h_{2}-6h_{1}\)
\(-8h_{4}-16h_{3}-12h_{2}-6h_{1}\)
\(-6g_{-19}-6g_{-20}\)
\(12g_{23}\)
\(-6g_{10}+6g_{9}\)
\(-12g_{-11}+12g_{-12}\)
\(36g_{-24}\)
\(36g_{-1}\)
Semisimple subalgebra component.
\(g_{4}-g_{3}\)
\(2h_{4}+2h_{3}\)
\(g_{-3}-g_{-4}\)
\(g_{18}\)
\(-g_{-2}\)
\(-g_{15}\)
\(g_{22}\)
\(-g_{-6}\)
\(-g_{12}-2g_{11}\)
\(-g_{7}\)
\(g_{21}\)
\(2g_{-9}+g_{-10}\)
\(-3g_{8}\)
\(2g_{-14}\)
\(-g_{4}-g_{3}\)
\(2g_{20}-g_{19}\)
\(-3g_{-13}\)
\(6g_{5}\)
\(-2g_{16}\)
\(-2g_{-17}\)
\(-2h_{4}+2h_{3}\)
\(-3g_{17}\)
\(-6g_{-16}\)
\(-2g_{-5}\)
\(2g_{13}\)
\(2g_{-19}-4g_{-20}\)
\(3g_{-3}+3g_{-4}\)
\(-6g_{14}\)
\(-2g_{-8}\)
\(2g_{10}+4g_{9}\)
\(6g_{-21}\)
\(-6g_{-7}\)
\(4g_{-11}+2g_{-12}\)
\(6g_{6}\)
\(-12g_{-22}\)
\(-6g_{-15}\)
\(-12g_{2}\)
\(-12g_{-18}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-3\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(3\omega_{1}-2\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+4\omega_{3}\)
\(\omega_{1}+2\omega_{3}\)
\(2\omega_{1}-\omega_{2}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(4\omega_{3}\)
\(2\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+4\omega_{3}\)
\(2\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(\omega_{1}-4\omega_{3}\)
\(\omega_{1}-\omega_{2}+4\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+2\omega_{3}\)
\(0\)
\(2\omega_{1}-\omega_{2}-2\omega_{3}\)
\(-\omega_{1}+\omega_{2}-4\omega_{3}\)
\(-\omega_{1}+4\omega_{3}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-2\omega_{3}\)
\(2\omega_{1}-\omega_{2}-4\omega_{3}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-2\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-4\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-4\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-3\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(3\omega_{1}-2\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+4\omega_{3}\)
\(\omega_{1}+2\omega_{3}\)
\(2\omega_{1}-\omega_{2}+4\omega_{3}\)
\(-\omega_{1}+\omega_{2}+2\omega_{3}\)
\(\omega_{1}\)
\(4\omega_{3}\)
\(2\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+4\omega_{3}\)
\(2\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(\omega_{1}-4\omega_{3}\)
\(\omega_{1}-\omega_{2}+4\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+2\omega_{3}\)
\(0\)
\(2\omega_{1}-\omega_{2}-2\omega_{3}\)
\(-\omega_{1}+\omega_{2}-4\omega_{3}\)
\(-\omega_{1}+4\omega_{3}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-2\omega_{3}\)
\(2\omega_{1}-\omega_{2}-4\omega_{3}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-2\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-4\omega_{3}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}-4\omega_{3}\)
\(-\omega_{1}-4\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{3\omega_{1}-\omega_{2}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus 2M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{1}+4\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+4\omega_{3}}
\oplus M_{4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+4\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{1}+4\omega_{3}}
\oplus M_{\omega_{1}+2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+2\omega_{3}}
\oplus M_{2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}
\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}
\oplus M_{-2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}-2\omega_{3}}
\oplus M_{\omega_{1}-4\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-4\omega_{3}}
\oplus M_{-4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-4\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{1}-4\omega_{3}}\)
Isotypic character\(\displaystyle M_{3\omega_{1}-\omega_{2}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{3\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus 2M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{1}+4\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+4\omega_{3}}
\oplus M_{4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+4\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{1}+4\omega_{3}}
\oplus M_{\omega_{1}+2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+2\omega_{3}}
\oplus M_{2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}
\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}
\oplus M_{-2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}-2\omega_{3}}
\oplus M_{\omega_{1}-4\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-4\omega_{3}}
\oplus M_{-4\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-4\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{1}-4\omega_{3}}\)

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 818444 arithmetic operations while solving the Serre relations polynomial system.