Subalgebra \(A^{8}_1+A^{4}_1\) ↪ \(F^{1}_4\)
24 out of 59
Computations done by the calculator project.

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

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 2879078 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_{-10}+g_{-15}, 2g_{15}+2g_{10},
x_{5} g_{-2}+x_{6} g_{-5}+x_{7} g_{-6}+x_{8} g_{-8}+x_{9} g_{-9}+x_{10} g_{-11}, x_{20} g_{11}+x_{19} g_{9}+x_{18} g_{8}+x_{17} g_{6}+x_{16} g_{5}+x_{15} g_{2})
h: (4, 8, 12, 8), e = combination of g_{10} g_{12} g_{13} g_{15} , f= combination of g_{-10} g_{-12} g_{-13} g_{-15} h: (2, 4, 4, 0), e = combination of g_{2} g_{5} g_{6} g_{8} g_{9} g_{11} , f= combination of g_{-2} g_{-5} g_{-6} g_{-8} g_{-9} g_{-11} Positive weight subsystem: 2 vectors: (1, 0), (0, 1)
Symmetric Cartan default scale: \begin{pmatrix}
2 & 0\\
0 & 2\\
\end{pmatrix}Character ambient Lie algebra: 2V_{4\omega_{1}+2\omega_{2}}+V_{4\omega_{2}}+2V_{2\omega_{1}+2\omega_{2}}+3V_{4\omega_{1}}+6V_{2\omega_{2}}+4V_{2\omega_{1}}+2V_{4\omega_{1}-2\omega_{2}}+2V_{-2\omega_{1}+2\omega_{2}}+8V_{0}+2V_{2\omega_{1}-2\omega_{2}}+2V_{-4\omega_{1}+2\omega_{2}}+4V_{-2\omega_{1}}+6V_{-2\omega_{2}}+3V_{-4\omega_{1}}+2V_{-2\omega_{1}-2\omega_{2}}+V_{-4\omega_{2}}+2V_{-4\omega_{1}-2\omega_{2}}
A necessary system to realize the candidate subalgebra.
x_{18} -x_{15} = 0
x_{20} +x_{17} = 0
x_{8} -x_{5} = 0
x_{10} +x_{7} = 0
x_{10} x_{20} +x_{9} x_{19} +2x_{8} x_{18} +2x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} -4= 0
x_{10} x_{19} +2x_{8} x_{17} +x_{6} x_{15} = 0
x_{10} x_{18} +x_{9} x_{17} -x_{8} x_{16} -x_{7} x_{15} = 0
x_{9} x_{20} +2x_{7} x_{18} +x_{5} x_{16} = 0
x_{10} x_{20} +2x_{8} x_{18} +x_{6} x_{16} -2= 0
x_{8} x_{20} +x_{7} x_{19} -x_{6} x_{18} -x_{5} x_{17} = 0
x_{10} x_{20} +x_{9} x_{19} +x_{8} x_{18} +x_{7} x_{17} -2= 0
The above system after transformation.
x_{18} -x_{15} = 0
x_{20} +x_{17} = 0
x_{8} -x_{5} = 0
x_{10} +x_{7} = 0
x_{10} x_{20} +x_{9} x_{19} +2x_{8} x_{18} +2x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} -4= 0
x_{10} x_{19} +2x_{8} x_{17} +x_{6} x_{15} = 0
x_{10} x_{18} +x_{9} x_{17} -x_{8} x_{16} -x_{7} x_{15} = 0
x_{9} x_{20} +2x_{7} x_{18} +x_{5} x_{16} = 0
x_{10} x_{20} +2x_{8} x_{18} +x_{6} x_{16} -2= 0
x_{8} x_{20} +x_{7} x_{19} -x_{6} x_{18} -x_{5} x_{17} = 0
x_{10} x_{20} +x_{9} x_{19} +x_{8} x_{18} +x_{7} x_{17} -2= 0
For the calculator:
(DynkinType =A^{8}_1+A^{4}_1; ElementsCartan =((4, 8, 12, 8), (2, 4, 4, 0)); generators =(g_{-10}+g_{-15}, 2g_{15}+2g_{10}, x_{5} g_{-2}+x_{6} g_{-5}+x_{7} g_{-6}+x_{8} g_{-8}+x_{9} g_{-9}+x_{10} g_{-11}, x_{20} g_{11}+x_{19} g_{9}+x_{18} g_{8}+x_{17} g_{6}+x_{16} g_{5}+x_{15} g_{2}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{18} -x_{15} , x_{20} +x_{17} , x_{8} -x_{5} , x_{10} +x_{7} , x_{10} x_{20} +x_{9} x_{19} +2x_{8} x_{18} +2x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} -4, x_{10} x_{19} +2x_{8} x_{17} +x_{6} x_{15} , x_{10} x_{18} +x_{9} x_{17} -x_{8} x_{16} -x_{7} x_{15} , x_{9} x_{20} +2x_{7} x_{18} +x_{5} x_{16} , x_{10} x_{20} +2x_{8} x_{18} +x_{6} x_{16} -2, x_{8} x_{20} +x_{7} x_{19} -x_{6} x_{18} -x_{5} x_{17} , x_{10} x_{20} +x_{9} x_{19} +x_{8} x_{18} +x_{7} x_{17} -2 )