Subalgebra \(A^{2}_2+A^{4}_1\) ↪ \(F^{1}_4\)
48 out of 59
Computations done by the calculator project.

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

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 2776 arithmetic operations while solving the Serre relations polynomial system.