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

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

Semisimple subalgebra: W_{3}+W_{4}
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 2776 arithmetic operations while solving the Serre relations polynomial system.