Subalgebra \(A^{1}_3+A^{2}_1\) ↪ \(F^{1}_4\)
55 out of 59
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{1}_3+A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) .
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: \(\displaystyle A^{1}_3\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -1, 0, 0): 2, \(\displaystyle A^{2}_1\): (0, 0, 0, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 18.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-4}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -1 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -1 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+2\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\)
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.\(g_{22}\)\(g_{21}\)\(g_{7}\)\(g_{4}\)\(g_{16}\)
weight\(\omega_{1}+\omega_{3}\)\(\omega_{1}+\omega_{4}\)\(\omega_{3}+\omega_{4}\)\(2\omega_{4}\)\(\omega_{2}+2\omega_{4}\)
Isotypic module decomposition over primal subalgebra (total 5 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}+\omega_{3}} \) → (1, 0, 1, 0)\(\displaystyle V_{\omega_{1}+\omega_{4}} \) → (1, 0, 0, 1)\(\displaystyle V_{\omega_{3}+\omega_{4}} \) → (0, 0, 1, 1)\(\displaystyle V_{2\omega_{4}} \) → (0, 0, 0, 2)\(\displaystyle V_{\omega_{2}+2\omega_{4}} \) → (0, 1, 0, 2)
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.
\(-g_{22}\)
\(g_{-5}\)
\(-g_{23}\)
\(-g_{-2}\)
\(g_{-1}\)
\(-g_{24}\)
\(-h_{2}\)
\(h_{1}\)
\(2h_{4}+4h_{3}+3h_{2}+2h_{1}\)
\(-g_{1}\)
\(2g_{2}\)
\(g_{-24}\)
\(-g_{-23}\)
\(g_{5}\)
\(g_{-22}\)
\(g_{21}\)
\(-g_{-8}\)
\(g_{19}\)
\(g_{-6}\)
\(-g_{-12}\)
\(-g_{-3}\)
\(g_{-10}\)
\(-g_{-7}\)
\(g_{7}\)
\(g_{10}\)
\(-g_{3}\)
\(g_{12}\)
\(-g_{6}\)
\(g_{-19}\)
\(-g_{8}\)
\(-g_{-21}\)
Semisimple subalgebra component.
\(-g_{4}\)
\(2h_{4}\)
\(2g_{-4}\)
\(g_{16}\)
\(g_{18}\)
\(g_{13}\)
\(-g_{-14}\)
\(g_{20}\)
\(g_{15}\)
\(-2g_{9}\)
\(g_{-11}\)
\(-g_{-17}\)
\(g_{17}\)
\(-2g_{11}\)
\(-g_{-9}\)
\(g_{-15}\)
\(2g_{-20}\)
\(-2g_{14}\)
\(-g_{-13}\)
\(-2g_{-18}\)
\(2g_{-16}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{3}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)		\(\omega_{2}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}+2\omega_{4}\)
\(\omega_{2}\)
\(-\omega_{1}+\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}\)
\(-\omega_{2}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{3}-2\omega_{4}\)
\(\omega_{1}-\omega_{3}-2\omega_{4}\)
\(-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}-2\omega_{4}\)
\(-\omega_{2}-2\omega_{4}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{3}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)		\(\omega_{2}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}+2\omega_{4}\)
\(\omega_{2}\)
\(-\omega_{1}+\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}\)
\(-\omega_{2}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{3}-2\omega_{4}\)
\(\omega_{1}-\omega_{3}-2\omega_{4}\)
\(-\omega_{2}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}-2\omega_{4}\)
\(-\omega_{2}-2\omega_{4}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}}
\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus 3M_{0}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{3}}\)		\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{-\omega_{3}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)		\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}
\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)\(\displaystyle M_{\omega_{1}-\omega_{2}+\omega_{3}+2\omega_{4}}\oplus M_{\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{1}-\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}
\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}}\oplus M_{\omega_{2}-2\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{3}-2\omega_{4}}\oplus M_{\omega_{1}-\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{2}-2\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}-2\omega_{4}}\)
Isotypic character\(\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}}
\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus 3M_{0}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{3}}\)		\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{-\omega_{3}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)		\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}
\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)\(\displaystyle M_{\omega_{1}-\omega_{2}+\omega_{3}+2\omega_{4}}\oplus M_{\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{1}-\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}
\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}}\oplus M_{\omega_{2}-2\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{3}-2\omega_{4}}\oplus M_{\omega_{1}-\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{2}-2\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}-2\omega_{4}}\)

Semisimple subalgebra: W_{1}+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 527 arithmetic operations while solving the Serre relations polynomial system.