Subalgebra \(A^{3}_2\) ↪ \(F^{1}_4\)
36 out of 59
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) .
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: \(\displaystyle A^{3}_2\): (3, 6, 8, 4): 6, (0, -3, -4, -2): 6
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \(\displaystyle g_{-14}+g_{-20}+g_{-22}\), \(\displaystyle -g_{16}+g_{10}+g_{9}\)
Positive simple generators: \(\displaystyle g_{22}+g_{20}+g_{14}\), \(\displaystyle g_{-2}+g_{-9}+g_{-10}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1/3\\ -1/3 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\ -3 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{2}}\oplus V_{3\omega_{1}}\oplus 4V_{\omega_{1}+\omega_{2}}\)
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 6) ; the vectors are over the primal subalgebra.\(-g_{12}+g_{11}+g_{5}\)\(g_{15}\)\(g_{18}-1/2g_{11}+1/2g_{5}\)\(g_{8}\)\(g_{24}\)\(g_{1}\)
weight\(\omega_{1}+\omega_{2}\)\(\omega_{1}+\omega_{2}\)\(\omega_{1}+\omega_{2}\)\(\omega_{1}+\omega_{2}\)\(3\omega_{1}\)\(3\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 4 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}+\omega_{2}} \) → (1, 1)\(\displaystyle V_{3\omega_{1}} \) → (3, 0)\(\displaystyle V_{3\omega_{2}} \) → (0, 3)
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.
\(g_{12}-g_{11}-g_{5}\)
\(g_{-2}+g_{-9}+g_{-10}\)
\(-g_{22}-g_{20}-g_{14}\)
\(2h_{4}+4h_{3}+3h_{2}\)
\(4h_{4}+8h_{3}+6h_{2}+3h_{1}\)
\(g_{-14}+g_{-20}+g_{-22}\)
\(2g_{16}-2g_{10}-2g_{9}\)
\(-g_{-11}+g_{-12}+g_{-18}\)
\(g_{8}\)
\(-g_{-6}\)
\(g_{21}+g_{17}\)
\(g_{4}-g_{3}\)
\(g_{4}-g_{3}-g_{-3}-g_{-4}\)
\(g_{-17}-g_{-21}\)
\(2g_{13}\)
\(-g_{-15}\)
\(g_{15}\)
\(g_{-13}\)
\(-g_{21}\)
\(-g_{4}+g_{-3}-g_{-4}\)
\(g_{3}+g_{-3}\)
\(g_{-21}\)
\(-2g_{13}-2g_{6}\)
\(-g_{-8}+g_{-15}\)
\(g_{18}-1/2g_{11}+1/2g_{5}\)
\(-1/2g_{-2}+1/2g_{-9}-g_{-16}\)
\(-3/2g_{22}-1/2g_{20}-1/2g_{19}+1/2g_{14}\)
\(1/2g_{7}+2h_{4}+3h_{3}+3/2h_{2}+g_{-7}\)
\(-1/2g_{7}+4h_{4}+6h_{3}+3h_{2}+3/2h_{1}+1/2g_{-7}\)
\(-1/2g_{-14}-1/2g_{-19}+1/2g_{-20}+3/2g_{-22}\)
\(3g_{16}-2g_{10}-g_{9}+2g_{2}\)
\(g_{-5}-1/2g_{-11}+g_{-12}+3/2g_{-18}\)
\(g_{24}\)
\(g_{18}+g_{11}-g_{5}\)
\(-2g_{-2}+2g_{-9}+2g_{-16}\)
\(g_{20}+g_{19}-g_{14}\)
\(-6g_{-23}\)
\(2g_{7}-4h_{4}-2g_{-7}\)
\(6g_{-14}+6g_{-19}-6g_{-20}\)
\(4g_{10}-4g_{9}-4g_{2}\)
\(-12g_{-5}-12g_{-11}-12g_{-12}\)
\(-36g_{-1}\)
\(g_{1}\)
\(g_{18}-g_{12}-g_{11}\)
\(g_{-9}-g_{-10}-g_{-16}\)
\(-2g_{22}+2g_{20}-2g_{19}\)
\(-g_{7}+2h_{3}+g_{-7}\)
\(-6g_{23}\)
\(-2g_{-19}-2g_{-20}+2g_{-22}\)
\(-2g_{16}-2g_{9}+2g_{2}\)
\(-2g_{-5}+2g_{-11}+2g_{-18}\)
\(-6g_{-24}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(3\omega_{1}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+3\omega_{2}\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{2}\)		\(3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(3\omega_{1}-3\omega_{2}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(3\omega_{1}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+3\omega_{2}\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{2}\)		\(3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(3\omega_{1}-3\omega_{2}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{3\omega_{1}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+3\omega_{2}}
\oplus M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{2}}\)		\(\displaystyle M_{3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{0}\oplus M_{3\omega_{1}-3\omega_{2}}
\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}}\)
Isotypic character\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle 3M_{\omega_{1}+\omega_{2}}\oplus 3M_{-\omega_{1}+2\omega_{2}}\oplus 3M_{2\omega_{1}-\omega_{2}}\oplus 6M_{0}\oplus 3M_{-2\omega_{1}+\omega_{2}}
\oplus 3M_{\omega_{1}-2\omega_{2}}\oplus 3M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{3\omega_{1}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+3\omega_{2}}
\oplus M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{2}}\)		\(\displaystyle M_{3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{0}\oplus M_{3\omega_{1}-3\omega_{2}}
\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}}\)

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 4426514 arithmetic operations while solving the Serre relations polynomial system.
The total number of arithmetic operations I needed to solve the Serre relations polynomial system was larger than 1 000 000. I am printing out the Serre relations system for you: maybe that can help improve the polynomial system algorithms.
Subalgebra realized.
2*2 (unknown) gens:
(
g_{-14}+g_{-20}+g_{-22}, g_{22}+g_{20}+g_{14},
x_{7} g_{16}+x_{8} g_{13}+x_{9} g_{10}+x_{10} g_{9}+x_{11} g_{6}+x_{12} g_{2}, x_{24} g_{-2}+x_{23} g_{-6}+x_{22} g_{-9}+x_{21} g_{-10}+x_{20} g_{-13}+x_{19} g_{-16})
h: (3, 6, 8, 4), e = combination of g_{14} g_{17} g_{19} g_{20} g_{21} g_{22} , f= combination of g_{-14} g_{-17} g_{-19} g_{-20} g_{-21} g_{-22} h: (0, -3, -4, -2), e = combination of g_{-16} g_{-13} g_{-10} g_{-9} g_{-6} g_{-2} , f= combination of g_{16} g_{13} g_{10} g_{9} g_{6} g_{2} Positive weight subsystem: 3 vectors: (1, 0), (0, 1), (1, 1)
Symmetric Cartan default scale: \begin{pmatrix}
2 & -1\\
-1 & 2\\
\end{pmatrix}Character ambient Lie algebra: V_{3\omega_{2}}+V_{3\omega_{1}}+6V_{\omega_{1}+\omega_{2}}+6V_{-\omega_{1}+2\omega_{2}}+6V_{2\omega_{1}-\omega_{2}}+V_{-3\omega_{1}+3\omega_{2}}+10V_{0}+V_{3\omega_{1}-3\omega_{2}}+6V_{-2\omega_{1}+\omega_{2}}+6V_{\omega_{1}-2\omega_{2}}+6V_{-\omega_{1}-\omega_{2}}+V_{-3\omega_{1}}+V_{-3\omega_{2}}
A necessary system to realize the candidate subalgebra.
x_{24} -x_{22} -x_{19} = 0
x_{12} -x_{10} -x_{7} = 0
x_{12} x_{24} +2x_{11} x_{23} +x_{10} x_{22} +2x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} -3= 0
x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} -2= 0
x_{9} x_{21} +x_{8} x_{20} +x_{7} x_{19} -1= 0
x_{11} x_{21} +x_{10} x_{20} -x_{8} x_{19} = 0
x_{12} x_{21} -x_{11} x_{20} +x_{9} x_{19} = 0
x_{9} x_{23} +x_{8} x_{22} -x_{7} x_{20} = 0
x_{12} x_{23} -x_{11} x_{22} -x_{9} x_{20} = 0
x_{9} x_{24} -x_{8} x_{23} +x_{7} x_{21} = 0
x_{11} x_{24} -x_{10} x_{23} -x_{8} x_{21} = 0
x_{22} x_{24} +x_{19} x_{24} +x_{23}^{2}-x_{19} x_{22} -x_{21}^{2}-x_{20}^{2}= 0
x_{10} x_{12} +x_{7} x_{12} +x_{11}^{2}-x_{7} x_{10} -x_{9}^{2}-x_{8}^{2}= 0
The above system after transformation.
x_{24} -x_{22} -x_{19} = 0
x_{12} -x_{10} -x_{7} = 0
x_{12} x_{24} +2x_{11} x_{23} +x_{10} x_{22} +2x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} -3= 0
x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} -2= 0
x_{9} x_{21} +x_{8} x_{20} +x_{7} x_{19} -1= 0
x_{11} x_{21} +x_{10} x_{20} -x_{8} x_{19} = 0
x_{12} x_{21} -x_{11} x_{20} +x_{9} x_{19} = 0
x_{9} x_{23} +x_{8} x_{22} -x_{7} x_{20} = 0
x_{12} x_{23} -x_{11} x_{22} -x_{9} x_{20} = 0
x_{9} x_{24} -x_{8} x_{23} +x_{7} x_{21} = 0
x_{11} x_{24} -x_{10} x_{23} -x_{8} x_{21} = 0
x_{22} x_{24} +x_{19} x_{24} +x_{23}^{2}-x_{19} x_{22} -x_{21}^{2}-x_{20}^{2}= 0
x_{10} x_{12} +x_{7} x_{12} +x_{11}^{2}-x_{7} x_{10} -x_{9}^{2}-x_{8}^{2}= 0
For the calculator:
(DynkinType =A^{3}_2; ElementsCartan =((3, 6, 8, 4), (0, -3, -4, -2)); generators =(g_{-14}+g_{-20}+g_{-22}, g_{22}+g_{20}+g_{14}, x_{7} g_{16}+x_{8} g_{13}+x_{9} g_{10}+x_{10} g_{9}+x_{11} g_{6}+x_{12} g_{2}, x_{24} g_{-2}+x_{23} g_{-6}+x_{22} g_{-9}+x_{21} g_{-10}+x_{20} g_{-13}+x_{19} g_{-16}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{24} -x_{22} -x_{19} , x_{12} -x_{10} -x_{7} , x_{12} x_{24} +2x_{11} x_{23} +x_{10} x_{22} +2x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} -3, x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} -2, x_{9} x_{21} +x_{8} x_{20} +x_{7} x_{19} -1, x_{11} x_{21} +x_{10} x_{20} -x_{8} x_{19} , x_{12} x_{21} -x_{11} x_{20} +x_{9} x_{19} , x_{9} x_{23} +x_{8} x_{22} -x_{7} x_{20} , x_{12} x_{23} -x_{11} x_{22} -x_{9} x_{20} , x_{9} x_{24} -x_{8} x_{23} +x_{7} x_{21} , x_{11} x_{24} -x_{10} x_{23} -x_{8} x_{21} , x_{22} x_{24} +x_{19} x_{24} +x_{23}^{2}-x_{19} x_{22} -x_{21}^{2}-x_{20}^{2}, x_{10} x_{12} +x_{7} x_{12} +x_{11}^{2}-x_{7} x_{10} -x_{9}^{2}-x_{8}^{2} )