Subalgebra \(F^{1}_4\) ↪ \(F^{1}_4\)
59 out of 59
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Subalgebra type: \(\displaystyle F^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_3\) .
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: \(\displaystyle F^{1}_4\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -2, -2, 0): 4, (0, 0, 0, -2): 4
Dimension of subalgebra generated by predefined or computed generators: 52.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{6}\), \(\displaystyle g_{4}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-6}\), \(\displaystyle g_{-4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -1 & 0\\ 0 & -1 & 1 & -1/2\\ 0 & 0 & -1/2 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -2 & 0\\ 0 & -2 & 4 & -2\\ 0 & 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}}\)
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 1) ; the vectors are over the primal subalgebra.\(g_{22}\)
weight\(\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 1 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}} \) → (1, 0, 0, 0)
Module label \(W_{1}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{22}\)
\(g_{-5}\)
\(-g_{-2}\)
\(g_{3}\)
\(-2g_{9}\)
\(-g_{7}\)
\(-2g_{11}\)
\(2g_{13}\)
\(2g_{-20}\)
\(2g_{15}\)
\(4g_{16}\)
\(-2g_{-17}\)
\(-2g_{19}\)
\(4g_{18}\)
\(2g_{-12}\)
\(-4g_{-14}\)
\(-2g_{21}\)
\(-2g_{-10}\)
\(2g_{-8}\)
\(-4g_{23}\)
\(2g_{-4}\)
\(-2g_{-6}\)
\(4g_{-1}\)
\(-4g_{24}\)
\(4h_{4}\)
\(-4h_{3}-4h_{2}\)
\(4h_{1}\)
\(8h_{4}+16h_{3}+12h_{2}+8h_{1}\)
\(2g_{6}\)
\(-4g_{4}\)
\(-4g_{1}\)
\(4g_{-24}\)
\(2g_{8}\)
\(-2g_{10}\)
\(-4g_{-23}\)
\(2g_{-21}\)
\(4g_{14}\)
\(-2g_{12}\)
\(4g_{-18}\)
\(-2g_{-19}\)
\(-4g_{17}\)
\(-4g_{-16}\)
\(-4g_{-15}\)
\(-8g_{20}\)
\(4g_{-13}\)
\(-8g_{-11}\)
\(4g_{-7}\)
\(8g_{-9}\)
\(4g_{-3}\)
\(8g_{2}\)
\(8g_{5}\)
\(8g_{-22}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}+2\omega_{3}\)
\(\omega_{4}\)
\(\omega_{2}-2\omega_{3}+2\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+2\omega_{4}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{1}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}-2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}+2\omega_{3}-2\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{3}+2\omega_{4}\)
\(-\omega_{2}+2\omega_{3}-\omega_{4}\)
\(-\omega_{1}+2\omega_{2}-2\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(0\)
\(\omega_{2}-2\omega_{3}+\omega_{4}\)
\(\omega_{3}-2\omega_{4}\)
\(\omega_{1}-2\omega_{2}+2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-2\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}+2\omega_{4}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{2}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{1}-2\omega_{4}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-2\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+2\omega_{3}-2\omega_{4}\)
\(-\omega_{4}\)
\(\omega_{2}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}+2\omega_{3}\)
\(\omega_{4}\)
\(\omega_{2}-2\omega_{3}+2\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+2\omega_{4}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{1}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}-2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}+2\omega_{3}-2\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{3}+2\omega_{4}\)
\(-\omega_{2}+2\omega_{3}-\omega_{4}\)
\(-\omega_{1}+2\omega_{2}-2\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(0\)
\(\omega_{2}-2\omega_{3}+\omega_{4}\)
\(\omega_{3}-2\omega_{4}\)
\(\omega_{1}-2\omega_{2}+2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-2\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}+2\omega_{4}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{2}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{1}-2\omega_{4}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-2\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+2\omega_{3}-2\omega_{4}\)
\(-\omega_{4}\)
\(\omega_{2}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{2}-2\omega_{3}+2\omega_{4}}\oplus M_{\omega_{1}-2\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}
\oplus M_{\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{3}}
\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{3}+\omega_{4}}\oplus M_{\omega_{2}-2\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}-\omega_{2}+2\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus 4M_{0}\oplus M_{\omega_{1}-\omega_{2}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}-\omega_{4}}
\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}-2\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}
\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{-\omega_{4}}
\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+2\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{3}-2\omega_{4}}
\oplus M_{\omega_{3}-2\omega_{4}}\oplus M_{\omega_{2}-2\omega_{4}}\oplus M_{\omega_{1}-2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{4}}\)
Isotypic character\(\displaystyle M_{\omega_{1}-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{2}-2\omega_{3}+2\omega_{4}}\oplus M_{\omega_{1}-2\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}
\oplus M_{\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{3}}
\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{3}+\omega_{4}}\oplus M_{\omega_{2}-2\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}-\omega_{2}+2\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus 4M_{0}\oplus M_{\omega_{1}-\omega_{2}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}-\omega_{4}}
\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+2\omega_{3}-2\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}
\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{-\omega_{4}}
\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+2\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{3}-2\omega_{4}}
\oplus M_{\omega_{3}-2\omega_{4}}\oplus M_{\omega_{2}-2\omega_{4}}\oplus M_{\omega_{1}-2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{4}}\)

Semisimple subalgebra: W_{1}
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 22979 arithmetic operations while solving the Serre relations polynomial system.