Subalgebra \(A^{9}_1+A^{3}_1\) ↪ \(F^{1}_4\)
25 out of 59
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{9}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) .
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: \(\displaystyle A^{9}_1\): (5, 10, 14, 8): 18, \(\displaystyle A^{3}_1\): (1, 0, 2, 0): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-12}+g_{-13}+g_{-14}\), \(\displaystyle g_{-1}-g_{-3}\)
Positive simple generators: \(\displaystyle g_{14}+2g_{13}+2g_{12}\), \(\displaystyle -g_{3}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0\\ 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0\\ 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}}
\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\)
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 7) ; the vectors are over the primal subalgebra.\(g_{14}\)\(g_{13}+g_{12}\)\(-g_{3}+g_{1}\)\(-g_{19}+4g_{18}\)\(g_{11}\)\(g_{24}\)\(g_{22}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{2}\)\(3\omega_{1}+\omega_{2}\)\(\omega_{1}+3\omega_{2}\)\(5\omega_{1}+\omega_{2}\)\(4\omega_{1}+2\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 7 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0)\(\displaystyle V_{2\omega_{2}} \) → (0, 2)\(\displaystyle V_{3\omega_{1}+\omega_{2}} \) → (3, 1)\(\displaystyle V_{\omega_{1}+3\omega_{2}} \) → (1, 3)\(\displaystyle V_{5\omega_{1}+\omega_{2}} \) → (5, 1)\(\displaystyle V_{4\omega_{1}+2\omega_{2}} \) → (4, 2)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-1/2g_{14}-g_{13}-g_{12}\)
\(4h_{4}+7h_{3}+5h_{2}+5/2h_{1}\)
\(g_{-12}+g_{-13}+g_{-14}\)
\(g_{14}\)
\(-2h_{3}-2h_{2}-h_{1}\)
\(-2g_{-14}\)
Semisimple subalgebra component.
\(g_{3}-g_{1}\)
\(2h_{3}+h_{1}\)
\(2g_{-1}-2g_{-3}\)
\(-g_{19}+4g_{18}\)
\(2g_{9}+g_{8}+3g_{7}\)
\(g_{17}+4g_{16}\)
\(3g_{-4}-4g_{-5}+2g_{-6}\)
\(-g_{6}+2g_{5}-3g_{4}\)
\(6g_{-16}+3g_{-17}\)
\(3g_{-7}+2g_{-8}+4g_{-9}\)
\(-6g_{-18}+3g_{-19}\)
\(g_{11}\)
\(-g_{-2}\)
\(g_{9}-g_{8}\)
\(g_{-5}+g_{-6}\)
\(-2g_{6}-2g_{5}\)
\(-2g_{-8}+2g_{-9}\)
\(-6g_{2}\)
\(-6g_{-11}\)
\(g_{24}\)
\(g_{19}+g_{18}\)
\(g_{23}\)
\(-2g_{9}-g_{8}+2g_{7}\)
\(-g_{17}+g_{16}\)
\(-3g_{-4}-6g_{-5}+3g_{-6}\)
\(g_{6}-2g_{5}-2g_{4}\)
\(-6g_{-16}+12g_{-17}\)
\(-3g_{-7}+3g_{-8}+6g_{-9}\)
\(30g_{-23}\)
\(6g_{-18}+12g_{-19}\)
\(-30g_{-24}\)
\(g_{22}\)
\(-g_{15}\)
\(-g_{21}\)
\(g_{3}+2g_{1}\)
\(-g_{13}+g_{12}\)
\(2g_{20}\)
\(3g_{-10}\)
\(2h_{3}-2h_{1}\)
\(2g_{10}\)
\(-6g_{-20}\)
\(-3g_{-12}+3g_{-13}\)
\(-4g_{-1}-2g_{-3}\)
\(-6g_{-21}\)
\(-6g_{-15}\)
\(-12g_{-22}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(\omega_{1}+3\omega_{2}\)
\(-\omega_{1}+3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(\omega_{1}-3\omega_{2}\)
\(-\omega_{1}-3\omega_{2}\)		\(5\omega_{1}+\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)		\(4\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-2\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(\omega_{1}+3\omega_{2}\)
\(-\omega_{1}+3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(\omega_{1}-3\omega_{2}\)
\(-\omega_{1}-3\omega_{2}\)		\(5\omega_{1}+\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)		\(4\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-2\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{1}+3\omega_{2}}\oplus M_{-\omega_{1}+3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}-3\omega_{2}}\oplus M_{-\omega_{1}-3\omega_{2}}\)		\(\displaystyle M_{5\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}
\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{1}+3\omega_{2}}\oplus M_{-\omega_{1}+3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}-3\omega_{2}}\oplus M_{-\omega_{1}-3\omega_{2}}\)		\(\displaystyle M_{5\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}
\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}\)

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 820990 arithmetic operations while solving the Serre relations polynomial system.