Subalgebra \(A^{28}_1+A^{8}_1\) ↪ \(F^{1}_4\)
29 out of 59
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Subalgebra type: \(\displaystyle A^{28}_1+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{28}_1\) .
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: \(\displaystyle A^{28}_1\): (10, 18, 24, 12): 56, \(\displaystyle A^{8}_1\): (0, 0, 4, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-9}+g_{-10}\), \(\displaystyle g_{-3}+g_{-4}\)
Positive simple generators: \(\displaystyle 6g_{10}+6g_{9}+10g_{1}\), \(\displaystyle 2g_{4}+2g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14 & 0\\ 0 & 1/4\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56 & 0\\ 0 & 16\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}+4\omega_{2}}\oplus V_{10\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
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 4) ; the vectors are over the primal subalgebra.\(g_{10}+g_{9}+5/3g_{1}\)\(g_{4}+g_{3}\)\(g_{24}\)\(g_{22}\)
weight\(2\omega_{1}\)\(2\omega_{2}\)\(10\omega_{1}\)\(6\omega_{1}+4\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 4 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0)\(\displaystyle V_{2\omega_{2}} \) → (0, 2)\(\displaystyle V_{10\omega_{1}} \) → (10, 0)\(\displaystyle V_{6\omega_{1}+4\omega_{2}} \) → (6, 4)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-3/5g_{10}-3/5g_{9}-g_{1}\)
\(6/5h_{4}+12/5h_{3}+9/5h_{2}+h_{1}\)
\(1/5g_{-1}+1/5g_{-9}+1/5g_{-10}\)
Semisimple subalgebra component.
\(-g_{4}-g_{3}\)
\(2h_{4}+2h_{3}\)
\(g_{-3}+g_{-4}\)
\(g_{24}\)
\(g_{23}\)
\(-g_{20}+g_{19}\)
\(2g_{12}+2g_{11}\)
\(2g_{10}+2g_{9}-6g_{1}\)
\(-4h_{4}-8h_{3}-6h_{2}+6h_{1}\)
\(18g_{-1}-10g_{-9}-10g_{-10}\)
\(28g_{-11}+28g_{-12}\)
\(-56g_{-19}+56g_{-20}\)
\(168g_{-23}\)
\(-168g_{-24}\)
\(g_{22}\)
\(-g_{18}\)
\(g_{21}\)
\(-g_{16}\)
\(-g_{15}\)
\(2g_{20}+g_{19}\)
\(-g_{7}\)
\(-g_{13}\)
\(-g_{12}+2g_{11}\)
\(3g_{17}\)
\(2g_{-2}\)
\(-g_{4}+g_{3}\)
\(-g_{10}+2g_{9}\)
\(3g_{8}\)
\(-6g_{14}\)
\(-2g_{-5}\)
\(2g_{-6}\)
\(2h_{4}-2h_{3}\)
\(3g_{6}\)
\(-6g_{5}\)
\(2g_{-14}\)
\(-2g_{-8}\)
\(-4g_{-9}+2g_{-10}\)
\(-3g_{-3}+3g_{-4}\)
\(-6g_{2}\)
\(2g_{-17}\)
\(4g_{-11}-2g_{-12}\)
\(-6g_{-13}\)
\(-6g_{-7}\)
\(-2g_{-19}-4g_{-20}\)
\(6g_{-15}\)
\(12g_{-16}\)
\(6g_{-21}\)
\(-12g_{-18}\)
\(-12g_{-22}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(6\omega_{1}+4\omega_{2}\)
\(4\omega_{1}+4\omega_{2}\)
\(6\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+4\omega_{2}\)
\(4\omega_{1}+2\omega_{2}\)
\(6\omega_{1}\)
\(4\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(6\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+4\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(6\omega_{1}-4\omega_{2}\)
\(-4\omega_{1}+4\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(4\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}+4\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(2\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}+2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{2}\)
\(-6\omega_{1}\)
\(-4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}-4\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(6\omega_{1}+4\omega_{2}\)
\(4\omega_{1}+4\omega_{2}\)
\(6\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+4\omega_{2}\)
\(4\omega_{1}+2\omega_{2}\)
\(6\omega_{1}\)
\(4\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(6\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+4\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(6\omega_{1}-4\omega_{2}\)
\(-4\omega_{1}+4\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(4\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}+4\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(2\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}+2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{2}\)
\(-6\omega_{1}\)
\(-4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-4\omega_{2}\)
\(-6\omega_{1}-4\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{6\omega_{1}+4\omega_{2}}\oplus M_{4\omega_{1}+4\omega_{2}}\oplus M_{6\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+4\omega_{2}}
\oplus M_{4\omega_{1}+2\omega_{2}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{6\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+4\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}\oplus M_{6\omega_{1}-4\omega_{2}}
\oplus M_{-4\omega_{1}+4\omega_{2}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{4\omega_{1}-4\omega_{2}}
\oplus M_{-6\omega_{1}+4\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{2}}\oplus M_{2\omega_{1}-4\omega_{2}}
\oplus M_{-6\omega_{1}+2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{2}}\oplus M_{-6\omega_{1}}
\oplus M_{-4\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{1}-4\omega_{2}}\oplus M_{-6\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-4\omega_{2}}
\oplus M_{-6\omega_{1}-4\omega_{2}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{6\omega_{1}+4\omega_{2}}\oplus M_{4\omega_{1}+4\omega_{2}}\oplus M_{6\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+4\omega_{2}}
\oplus M_{4\omega_{1}+2\omega_{2}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{6\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+4\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}\oplus M_{6\omega_{1}-4\omega_{2}}
\oplus M_{-4\omega_{1}+4\omega_{2}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{4\omega_{1}-4\omega_{2}}
\oplus M_{-6\omega_{1}+4\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{2}}\oplus M_{2\omega_{1}-4\omega_{2}}
\oplus M_{-6\omega_{1}+2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{2}}\oplus M_{-6\omega_{1}}
\oplus M_{-4\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{1}-4\omega_{2}}\oplus M_{-6\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-4\omega_{2}}
\oplus M_{-6\omega_{1}-4\omega_{2}}\)

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 682784 arithmetic operations while solving the Serre relations polynomial system.