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

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

Semisimple subalgebra: W_{1}+W_{3}
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 529 arithmetic operations while solving the Serre relations polynomial system.