Subalgebra A22+A21C15
89 out of 119
Computations done by the calculator project.

Subalgebra type: A22+A21 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A22 .
Centralizer: T2 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: C15
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (2, 0, -2, 0, -1)

Elements Cartan subalgebra scaled to act by two by components: A22: (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, A21: (0, 0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: g24, g2, g9
Positive simple generators: g24, g2, g9
Cartan symmetric matrix: (11/201/210001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (420240004)
Decomposition of ambient Lie algebra: 3V2ω32Vω2+ω32Vω1+ω3V2ω2Vω1+ω2V2ω12V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ω3+2ψ1+4ψ2Vω1+ω3+ψ1+4ψ2V2ω1+4ψ2Vω2+ω3+ψ1V2ω3Vω1+ω2Vω1+ω3ψ12V0V2ω24ψ2Vω2+ω3ψ14ψ2V2ω32ψ14ψ2
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. As the centralizer is well-chosen and the centralizer of our subalgebra is non-trivial, we may in addition split highest weight vectors with the same weight over the semisimple part over the centralizer (recall that the centralizer preserves the weights over the subalgebra and in particular acts on the highest weight vectors). Therefore we have chosen our highest weight vectors to be, in addition, weight vectors over the Cartan of the centralizer of the starting subalgebra. Their weight over the sum of the Cartans of the semisimple subalgebra and its centralizer is indicated in the third row. The weights corresponding to the Cartan of the centralizer are again indicated with the letter \omega. As there is no preferred way of chosing a basis of the Cartan of the centralizer (unlike the starting semisimple Lie algebra: there we have a preferred basis induced by the fundamental weights), our centralizer weights are simply given by the constant by which the k^th basis element of the Cartan of the centralizer acts on the highest weight vector. Here, we use the choice for basis of the Cartan of the centralizer given at the start of the page.

Highest vectors of representations (total 12) ; the vectors are over the primal subalgebra.h52h3+2h1h4g25g22g19g17g20g12g16g5g9g13
weight002ω1ω1+ω22ω2ω1+ω3ω1+ω3ω2+ω3ω2+ω32ω32ω32ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). 002ω1+4ψ2ω1+ω22ω24ψ2ω1+ω3ψ1ω1+ω3+ψ1+4ψ2ω2+ω3ψ14ψ2ω2+ω3+ψ12ω32ψ14ψ22ω32ω3+2ψ1+4ψ2
Isotypic module decomposition over primal subalgebra (total 11 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0, 0)V2ω1+4ψ2 → (2, 0, 0, 0, 4)Vω1+ω2 → (1, 1, 0, 0, 0)V2ω24ψ2 → (0, 2, 0, 0, -4)Vω1+ω3ψ1 → (1, 0, 1, -1, 0)Vω1+ω3+ψ1+4ψ2 → (1, 0, 1, 1, 4)Vω2+ω3ψ14ψ2 → (0, 1, 1, -1, -4)Vω2+ω3+ψ1 → (0, 1, 1, 1, 0)V2ω32ψ14ψ2 → (0, 0, 2, -2, -4)V2ω3 → (0, 0, 2, 0, 0)V2ω3+2ψ1+4ψ2 → (0, 0, 2, 2, 4)
Module label W1W2W3W4W5W6W7W8W9W10W11
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
h52h3+2h1
h4
g25
g1
2g23
g6
2g21
4g19
Semisimple subalgebra component.
g22
g2
g24
2h2
2h5+4h4+4h3+4h2+2h1
g24
2g2
g22
g19
g21
g6
2g23
g1
2g25
g17
g11
g10
g8
g18
g16
g20
g7
g14
g3
g15
g12
g12
g15
g3
g14
g7
g20
g16
g18
g8
g10
g11
g17
g5
g4
2g13
Semisimple subalgebra component.
g9
2h5+2h4
2g9
g13
g4
2g5
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above02ω1
ω2
2ω1+2ω2
ω1ω2
ω1
2ω2
ω1+ω2
ω1+2ω2
2ω1ω2
0
0
2ω1+ω2
ω12ω2
ω1ω2
2ω2
ω1
ω1+ω2
2ω12ω2
ω2
2ω1
ω1+ω3
ω1+ω2+ω3
ω1ω3
ω2+ω3
ω1+ω2ω3
ω2ω3
ω1+ω3
ω1+ω2+ω3
ω1ω3
ω2+ω3
ω1+ω2ω3
ω2ω3
ω2+ω3
ω1ω2+ω3
ω2ω3
ω1+ω3
ω1ω2ω3
ω1ω3
ω2+ω3
ω1ω2+ω3
ω2ω3
ω1+ω3
ω1ω2ω3
ω1ω3
2ω3
0
2ω3
2ω3
0
2ω3
2ω3
0
2ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer02ω1+4ψ2
ω2+4ψ2
2ω1+2ω2+4ψ2
ω1ω2+4ψ2
ω1+4ψ2
2ω2+4ψ2
ω1+ω2
ω1+2ω2
2ω1ω2
0
0
2ω1+ω2
ω12ω2
ω1ω2
2ω24ψ2
ω14ψ2
ω1+ω24ψ2
2ω12ω24ψ2
ω24ψ2
2ω14ψ2
ω1+ω3ψ1
ω1+ω2+ω3ψ1
ω1ω3ψ1
ω2+ω3ψ1
ω1+ω2ω3ψ1
ω2ω3ψ1
ω1+ω3+ψ1+4ψ2
ω1+ω2+ω3+ψ1+4ψ2
ω1ω3+ψ1+4ψ2
ω2+ω3+ψ1+4ψ2
ω1+ω2ω3+ψ1+4ψ2
ω2ω3+ψ1+4ψ2
ω2+ω3ψ14ψ2
ω1ω2+ω3ψ14ψ2
ω2ω3ψ14ψ2
ω1+ω3ψ14ψ2
ω1ω2ω3ψ14ψ2
ω1ω3ψ14ψ2
ω2+ω3+ψ1
ω1ω2+ω3+ψ1
ω2ω3+ψ1
ω1+ω3+ψ1
ω1ω2ω3+ψ1
ω1ω3+ψ1
2ω32ψ14ψ2
2ψ14ψ2
2ω32ψ14ψ2
2ω3
0
2ω3
2ω3+2ψ1+4ψ2
2ψ1+4ψ2
2ω3+2ψ1+4ψ2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0M2ω1+4ψ2Mω2+4ψ2M2ω1+2ω2+4ψ2Mω1ω2+4ψ2Mω1+4ψ2M2ω2+4ψ2Mω1+ω2Mω1+2ω2M2ω1ω22M0M2ω1+ω2Mω12ω2Mω1ω2M2ω24ψ2Mω14ψ2Mω1+ω24ψ2M2ω12ω24ψ2Mω24ψ2M2ω14ψ2Mω1+ω3ψ1Mω1+ω2+ω3ψ1Mω2+ω3ψ1Mω1ω3ψ1Mω1+ω2ω3ψ1Mω2ω3ψ1Mω1+ω3+ψ1+4ψ2Mω1+ω2+ω3+ψ1+4ψ2Mω2+ω3+ψ1+4ψ2Mω1ω3+ψ1+4ψ2Mω1+ω2ω3+ψ1+4ψ2Mω2ω3+ψ1+4ψ2Mω2+ω3ψ14ψ2Mω1ω2+ω3ψ14ψ2Mω1+ω3ψ14ψ2Mω2ω3ψ14ψ2Mω1ω2ω3ψ14ψ2Mω1ω3ψ14ψ2Mω2+ω3+ψ1Mω1ω2+ω3+ψ1Mω1+ω3+ψ1Mω2ω3+ψ1Mω1ω2ω3+ψ1Mω1ω3+ψ1M2ω32ψ14ψ2M2ψ14ψ2M2ω32ψ14ψ2M2ω3M0M2ω3M2ω3+2ψ1+4ψ2M2ψ1+4ψ2M2ω3+2ψ1+4ψ2
Isotypic character2M0M2ω1+4ψ2Mω2+4ψ2M2ω1+2ω2+4ψ2Mω1ω2+4ψ2Mω1+4ψ2M2ω2+4ψ2Mω1+ω2Mω1+2ω2M2ω1ω22M0M2ω1+ω2Mω12ω2Mω1ω2M2ω24ψ2Mω14ψ2Mω1+ω24ψ2M2ω12ω24ψ2Mω24ψ2M2ω14ψ2Mω1+ω3ψ1Mω1+ω2+ω3ψ1Mω2+ω3ψ1Mω1ω3ψ1Mω1+ω2ω3ψ1Mω2ω3ψ1Mω1+ω3+ψ1+4ψ2Mω1+ω2+ω3+ψ1+4ψ2Mω2+ω3+ψ1+4ψ2Mω1ω3+ψ1+4ψ2Mω1+ω2ω3+ψ1+4ψ2Mω2ω3+ψ1+4ψ2Mω2+ω3ψ14ψ2Mω1ω2+ω3ψ14ψ2Mω1+ω3ψ14ψ2Mω2ω3ψ14ψ2Mω1ω2ω3ψ14ψ2Mω1ω3ψ14ψ2Mω2+ω3+ψ1Mω1ω2+ω3+ψ1Mω1+ω3+ψ1Mω2ω3+ψ1Mω1ω2ω3+ψ1Mω1ω3+ψ1M2ω32ψ14ψ2M2ψ14ψ2M2ω32ψ14ψ2M2ω3M0M2ω3M2ω3+2ψ1+4ψ2M2ψ1+4ψ2M2ω3+2ψ1+4ψ2

Semisimple subalgebra: W_{3}+W_{10}
Centralizer extension: W_{1}

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.
Canvas not supported




Mouse position: (0.00, 0.00)
Selected index: -1
Coordinate center in screen coordinates:
(200.00, 300.00)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00, 0.00, 0.00)
(0.00, 1.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00): (333.33, 366.67)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (266.67, 433.33)
2: (0.00, 0.00, 1.00, 0.00, 0.00): (200.00, 300.00)
3: (0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
4: (0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 300.00)




Made total 13856855 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.
3*2 (unknown) gens:
(
g_{-24}, g_{24},
g_{2}, g_{-2},
x_{3} g_{-5}+x_{4} g_{-9}+x_{5} g_{-13}, x_{10} g_{13}+x_{9} g_{9}+x_{8} g_{5})

Unknown splitting cartan of centralizer.
x_{15} h_{5}+x_{14} h_{4}+x_{13} h_{3}+x_{12} h_{2}+x_{11} h_{1}, x_{20} h_{5}+x_{19} h_{4}+x_{18} h_{3}+x_{17} h_{2}+x_{16} h_{1}
h: (2, 4, 4, 4, 2), e = combination of g_{24} , f= combination of g_{-24} h: (0, -2, 0, 0, 0), e = combination of g_{-2} , f= combination of g_{2} h: (0, 0, 0, 2, 2), e = combination of g_{5} g_{9} g_{13} , f= combination of g_{-5} g_{-9} g_{-13} Positive weight subsystem: 4 vectors: (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0)
Symmetric Cartan default scale: \begin{pmatrix}
2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}Character ambient Lie algebra: 3V_{2\omega_{3}}+2V_{\omega_{2}+\omega_{3}}+2V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+2V_{-\omega_{1}+\omega_{2}+\omega_{3}}+2V_{\omega_{1}-\omega_{2}+\omega_{3}}+V_{-\omega_{1}+2\omega_{2}}+V_{\omega_{2}}+V_{\omega_{1}}+V_{2\omega_{1}-\omega_{2}}+2V_{-\omega_{1}+\omega_{3}}+2V_{-\omega_{2}+\omega_{3}}+V_{-2\omega_{1}+2\omega_{2}}+V_{-\omega_{1}+\omega_{2}}+7V_{0}+V_{\omega_{1}-\omega_{2}}+V_{2\omega_{1}-2\omega_{2}}+2V_{\omega_{2}-\omega_{3}}+2V_{\omega_{1}-\omega_{3}}+V_{-2\omega_{1}+\omega_{2}}+V_{-\omega_{1}}+V_{-\omega_{2}}+V_{\omega_{1}-2\omega_{2}}+2V_{-\omega_{1}+\omega_{2}-\omega_{3}}+2V_{\omega_{1}-\omega_{2}-\omega_{3}}+V_{-2\omega_{1}}+V_{-\omega_{1}-\omega_{2}}+V_{-2\omega_{2}}+2V_{-\omega_{1}-\omega_{3}}+2V_{-\omega_{2}-\omega_{3}}+3V_{-2\omega_{3}}
A necessary system to realize the candidate subalgebra.
x_{14}^{2}x_{20}^{2}x_{21} -2x_{13} x_{14} x_{20}^{2}x_{21} +2x_{13}^{2}x_{20}^{2}x_{21} -2x_{12} x_{13} x_{20}^{2}x_{21}
+2x_{12}^{2}x_{20}^{2}x_{21} -2x_{11} x_{12} x_{20}^{2}x_{21} +2x_{11}^{2}x_{20}^{2}x_{21} -2x_{14} x_{15} x_{19} x_{20} x_{21}
+2x_{13} x_{15} x_{19} x_{20} x_{21} +x_{13} x_{14} x_{19} x_{20} x_{21} -2x_{13}^{2}x_{19} x_{20} x_{21}
+2x_{12} x_{13} x_{19} x_{20} x_{21} -2x_{12}^{2}x_{19} x_{20} x_{21} +2x_{11} x_{12} x_{19} x_{20} x_{21}
-2x_{11}^{2}x_{19} x_{20} x_{21} +2x_{14} x_{15} x_{18} x_{20} x_{21} -4x_{13} x_{15} x_{18} x_{20} x_{21}
+2x_{12} x_{15} x_{18} x_{20} x_{21} -x_{14}^{2}x_{18} x_{20} x_{21} +2x_{13} x_{14} x_{18} x_{20} x_{21}
-x_{12} x_{14} x_{18} x_{20} x_{21} +2x_{13} x_{15} x_{17} x_{20} x_{21} -4x_{12} x_{15} x_{17} x_{20} x_{21}
+2x_{11} x_{15} x_{17} x_{20} x_{21} -x_{13} x_{14} x_{17} x_{20} x_{21} +2x_{12} x_{14} x_{17} x_{20} x_{21}
-x_{11} x_{14} x_{17} x_{20} x_{21} +2x_{12} x_{15} x_{16} x_{20} x_{21} -4x_{11} x_{15} x_{16} x_{20} x_{21}
-x_{12} x_{14} x_{16} x_{20} x_{21} +2x_{11} x_{14} x_{16} x_{20} x_{21} +x_{15}^{2}x_{19}^{2}x_{21}
-x_{13} x_{15} x_{19}^{2}x_{21} +3/4x_{13}^{2}x_{19}^{2}x_{21} -x_{12} x_{13} x_{19}^{2}x_{21} +x_{12}^{2}x_{19}^{2}x_{21}
-x_{11} x_{12} x_{19}^{2}x_{21} +x_{11}^{2}x_{19}^{2}x_{21} -2x_{15}^{2}x_{18} x_{19} x_{21} +x_{14} x_{15} x_{18} x_{19} x_{21}
+2x_{13} x_{15} x_{18} x_{19} x_{21} -x_{12} x_{15} x_{18} x_{19} x_{21} -3/2x_{13} x_{14} x_{18} x_{19} x_{21}
+x_{12} x_{14} x_{18} x_{19} x_{21} +1/2x_{12} x_{13} x_{18} x_{19} x_{21} -x_{12}^{2}x_{18} x_{19} x_{21}
+x_{11} x_{12} x_{18} x_{19} x_{21} -x_{11}^{2}x_{18} x_{19} x_{21} -x_{13} x_{15} x_{17} x_{19} x_{21}
+2x_{12} x_{15} x_{17} x_{19} x_{21} -x_{11} x_{15} x_{17} x_{19} x_{21} +x_{13} x_{14} x_{17} x_{19} x_{21}
-2x_{12} x_{14} x_{17} x_{19} x_{21} +x_{11} x_{14} x_{17} x_{19} x_{21} -1/2x_{13}^{2}x_{17} x_{19} x_{21}
+x_{12} x_{13} x_{17} x_{19} x_{21} -1/2x_{11} x_{13} x_{17} x_{19} x_{21} -x_{12} x_{15} x_{16} x_{19} x_{21}
+2x_{11} x_{15} x_{16} x_{19} x_{21} +x_{12} x_{14} x_{16} x_{19} x_{21} -2x_{11} x_{14} x_{16} x_{19} x_{21}
-1/2x_{12} x_{13} x_{16} x_{19} x_{21} +x_{11} x_{13} x_{16} x_{19} x_{21} +2x_{15}^{2}x_{18}^{2}x_{21}
-2x_{14} x_{15} x_{18}^{2}x_{21} +3/4x_{14}^{2}x_{18}^{2}x_{21} -1/2x_{12} x_{14} x_{18}^{2}x_{21} +3/4x_{12}^{2}x_{18}^{2}x_{21}
-x_{11} x_{12} x_{18}^{2}x_{21} +x_{11}^{2}x_{18}^{2}x_{21} -2x_{15}^{2}x_{17} x_{18} x_{21} +2x_{14} x_{15} x_{17} x_{18} x_{21}
-x_{14}^{2}x_{17} x_{18} x_{21} +1/2x_{13} x_{14} x_{17} x_{18} x_{21} +x_{12} x_{14} x_{17} x_{18} x_{21}
-1/2x_{11} x_{14} x_{17} x_{18} x_{21} -3/2x_{12} x_{13} x_{17} x_{18} x_{21} +x_{11} x_{13} x_{17} x_{18} x_{21}
+1/2x_{11} x_{12} x_{17} x_{18} x_{21} -x_{11}^{2}x_{17} x_{18} x_{21} -1/2x_{12} x_{14} x_{16} x_{18} x_{21}
+x_{11} x_{14} x_{16} x_{18} x_{21} +x_{12} x_{13} x_{16} x_{18} x_{21} -2x_{11} x_{13} x_{16} x_{18} x_{21}
-1/2x_{12}^{2}x_{16} x_{18} x_{21} +x_{11} x_{12} x_{16} x_{18} x_{21} +2x_{15}^{2}x_{17}^{2}x_{21} -2x_{14} x_{15} x_{17}^{2}x_{21}
+x_{14}^{2}x_{17}^{2}x_{21} -x_{13} x_{14} x_{17}^{2}x_{21} +3/4x_{13}^{2}x_{17}^{2}x_{21} -1/2x_{11} x_{13} x_{17}^{2}x_{21}
+3/4x_{11}^{2}x_{17}^{2}x_{21} -2x_{15}^{2}x_{16} x_{17} x_{21} +2x_{14} x_{15} x_{16} x_{17} x_{21}
-x_{14}^{2}x_{16} x_{17} x_{21} +x_{13} x_{14} x_{16} x_{17} x_{21} -x_{13}^{2}x_{16} x_{17} x_{21} +1/2x_{12} x_{13} x_{16} x_{17} x_{21}
+x_{11} x_{13} x_{16} x_{17} x_{21} -3/2x_{11} x_{12} x_{16} x_{17} x_{21} +2x_{15}^{2}x_{16}^{2}x_{21}
-2x_{14} x_{15} x_{16}^{2}x_{21} +x_{14}^{2}x_{16}^{2}x_{21} -x_{13} x_{14} x_{16}^{2}x_{21} +x_{13}^{2}x_{16}^{2}x_{21}
-x_{12} x_{13} x_{16}^{2}x_{21} +3/4x_{12}^{2}x_{16}^{2}x_{21} -1= 0
x_{12} = 0
x_{17} = 0
x_{13} -2x_{12} +x_{11} = 0
x_{18} -2x_{17} +x_{16} = 0
x_{5} x_{10} +2x_{4} x_{9} +x_{3} x_{8} -2= 0
x_{5} x_{9} +x_{4} x_{8} = 0
x_{4} x_{10} +x_{3} x_{9} = 0
x_{5} x_{10} +x_{4} x_{9} -1= 0
2x_{3} x_{15} -x_{3} x_{14} = 0
2x_{4} x_{15} -x_{4} x_{13} = 0
x_{5} x_{14} -x_{5} x_{13} = 0
2x_{8} x_{15} -x_{8} x_{14} = 0
2x_{9} x_{15} -x_{9} x_{13} = 0
x_{10} x_{14} -x_{10} x_{13} = 0
2x_{3} x_{20} -x_{3} x_{19} = 0
2x_{4} x_{20} -x_{4} x_{18} = 0
x_{5} x_{19} -x_{5} x_{18} = 0
2x_{8} x_{20} -x_{8} x_{19} = 0
2x_{9} x_{20} -x_{9} x_{18} = 0
x_{10} x_{19} -x_{10} x_{18} = 0
The above system after transformation.
x_{14}^{2}x_{20}^{2}x_{21} -2x_{13} x_{14} x_{20}^{2}x_{21} +2x_{13}^{2}x_{20}^{2}x_{21} -2x_{12} x_{13} x_{20}^{2}x_{21}
+2x_{12}^{2}x_{20}^{2}x_{21} -2x_{11} x_{12} x_{20}^{2}x_{21} +2x_{11}^{2}x_{20}^{2}x_{21} -2x_{14} x_{15} x_{19} x_{20} x_{21}
+2x_{13} x_{15} x_{19} x_{20} x_{21} +x_{13} x_{14} x_{19} x_{20} x_{21} -2x_{13}^{2}x_{19} x_{20} x_{21}
+2x_{12} x_{13} x_{19} x_{20} x_{21} -2x_{12}^{2}x_{19} x_{20} x_{21} +2x_{11} x_{12} x_{19} x_{20} x_{21}
-2x_{11}^{2}x_{19} x_{20} x_{21} +2x_{14} x_{15} x_{18} x_{20} x_{21} -4x_{13} x_{15} x_{18} x_{20} x_{21}
+2x_{12} x_{15} x_{18} x_{20} x_{21} -x_{14}^{2}x_{18} x_{20} x_{21} +2x_{13} x_{14} x_{18} x_{20} x_{21}
-x_{12} x_{14} x_{18} x_{20} x_{21} +2x_{13} x_{15} x_{17} x_{20} x_{21} -4x_{12} x_{15} x_{17} x_{20} x_{21}
+2x_{11} x_{15} x_{17} x_{20} x_{21} -x_{13} x_{14} x_{17} x_{20} x_{21} +2x_{12} x_{14} x_{17} x_{20} x_{21}
-x_{11} x_{14} x_{17} x_{20} x_{21} +2x_{12} x_{15} x_{16} x_{20} x_{21} -4x_{11} x_{15} x_{16} x_{20} x_{21}
-x_{12} x_{14} x_{16} x_{20} x_{21} +2x_{11} x_{14} x_{16} x_{20} x_{21} +x_{15}^{2}x_{19}^{2}x_{21}
-x_{13} x_{15} x_{19}^{2}x_{21} +3/4x_{13}^{2}x_{19}^{2}x_{21} -x_{12} x_{13} x_{19}^{2}x_{21} +x_{12}^{2}x_{19}^{2}x_{21}
-x_{11} x_{12} x_{19}^{2}x_{21} +x_{11}^{2}x_{19}^{2}x_{21} -2x_{15}^{2}x_{18} x_{19} x_{21} +x_{14} x_{15} x_{18} x_{19} x_{21}
+2x_{13} x_{15} x_{18} x_{19} x_{21} -x_{12} x_{15} x_{18} x_{19} x_{21} -3/2x_{13} x_{14} x_{18} x_{19} x_{21}
+x_{12} x_{14} x_{18} x_{19} x_{21} +1/2x_{12} x_{13} x_{18} x_{19} x_{21} -x_{12}^{2}x_{18} x_{19} x_{21}
+x_{11} x_{12} x_{18} x_{19} x_{21} -x_{11}^{2}x_{18} x_{19} x_{21} -x_{13} x_{15} x_{17} x_{19} x_{21}
+2x_{12} x_{15} x_{17} x_{19} x_{21} -x_{11} x_{15} x_{17} x_{19} x_{21} +x_{13} x_{14} x_{17} x_{19} x_{21}
-2x_{12} x_{14} x_{17} x_{19} x_{21} +x_{11} x_{14} x_{17} x_{19} x_{21} -1/2x_{13}^{2}x_{17} x_{19} x_{21}
+x_{12} x_{13} x_{17} x_{19} x_{21} -1/2x_{11} x_{13} x_{17} x_{19} x_{21} -x_{12} x_{15} x_{16} x_{19} x_{21}
+2x_{11} x_{15} x_{16} x_{19} x_{21} +x_{12} x_{14} x_{16} x_{19} x_{21} -2x_{11} x_{14} x_{16} x_{19} x_{21}
-1/2x_{12} x_{13} x_{16} x_{19} x_{21} +x_{11} x_{13} x_{16} x_{19} x_{21} +2x_{15}^{2}x_{18}^{2}x_{21}
-2x_{14} x_{15} x_{18}^{2}x_{21} +3/4x_{14}^{2}x_{18}^{2}x_{21} -1/2x_{12} x_{14} x_{18}^{2}x_{21} +3/4x_{12}^{2}x_{18}^{2}x_{21}
-x_{11} x_{12} x_{18}^{2}x_{21} +x_{11}^{2}x_{18}^{2}x_{21} -2x_{15}^{2}x_{17} x_{18} x_{21} +2x_{14} x_{15} x_{17} x_{18} x_{21}
-x_{14}^{2}x_{17} x_{18} x_{21} +1/2x_{13} x_{14} x_{17} x_{18} x_{21} +x_{12} x_{14} x_{17} x_{18} x_{21}
-1/2x_{11} x_{14} x_{17} x_{18} x_{21} -3/2x_{12} x_{13} x_{17} x_{18} x_{21} +x_{11} x_{13} x_{17} x_{18} x_{21}
+1/2x_{11} x_{12} x_{17} x_{18} x_{21} -x_{11}^{2}x_{17} x_{18} x_{21} -1/2x_{12} x_{14} x_{16} x_{18} x_{21}
+x_{11} x_{14} x_{16} x_{18} x_{21} +x_{12} x_{13} x_{16} x_{18} x_{21} -2x_{11} x_{13} x_{16} x_{18} x_{21}
-1/2x_{12}^{2}x_{16} x_{18} x_{21} +x_{11} x_{12} x_{16} x_{18} x_{21} +2x_{15}^{2}x_{17}^{2}x_{21} -2x_{14} x_{15} x_{17}^{2}x_{21}
+x_{14}^{2}x_{17}^{2}x_{21} -x_{13} x_{14} x_{17}^{2}x_{21} +3/4x_{13}^{2}x_{17}^{2}x_{21} -1/2x_{11} x_{13} x_{17}^{2}x_{21}
+3/4x_{11}^{2}x_{17}^{2}x_{21} -2x_{15}^{2}x_{16} x_{17} x_{21} +2x_{14} x_{15} x_{16} x_{17} x_{21}
-x_{14}^{2}x_{16} x_{17} x_{21} +x_{13} x_{14} x_{16} x_{17} x_{21} -x_{13}^{2}x_{16} x_{17} x_{21} +1/2x_{12} x_{13} x_{16} x_{17} x_{21}
+x_{11} x_{13} x_{16} x_{17} x_{21} -3/2x_{11} x_{12} x_{16} x_{17} x_{21} +2x_{15}^{2}x_{16}^{2}x_{21}
-2x_{14} x_{15} x_{16}^{2}x_{21} +x_{14}^{2}x_{16}^{2}x_{21} -x_{13} x_{14} x_{16}^{2}x_{21} +x_{13}^{2}x_{16}^{2}x_{21}
-x_{12} x_{13} x_{16}^{2}x_{21} +3/4x_{12}^{2}x_{16}^{2}x_{21} -1= 0
x_{12} = 0
x_{17} = 0
x_{13} -2x_{12} +x_{11} = 0
x_{18} -2x_{17} +x_{16} = 0
x_{5} x_{10} +2x_{4} x_{9} +x_{3} x_{8} -2= 0
x_{5} x_{9} +x_{4} x_{8} = 0
x_{4} x_{10} +x_{3} x_{9} = 0
x_{5} x_{10} +x_{4} x_{9} -1= 0
2x_{3} x_{15} -x_{3} x_{14} = 0
2x_{4} x_{15} -x_{4} x_{13} = 0
x_{5} x_{14} -x_{5} x_{13} = 0
2x_{8} x_{15} -x_{8} x_{14} = 0
2x_{9} x_{15} -x_{9} x_{13} = 0
x_{10} x_{14} -x_{10} x_{13} = 0
2x_{3} x_{20} -x_{3} x_{19} = 0
2x_{4} x_{20} -x_{4} x_{18} = 0
x_{5} x_{19} -x_{5} x_{18} = 0
2x_{8} x_{20} -x_{8} x_{19} = 0
2x_{9} x_{20} -x_{9} x_{18} = 0
x_{10} x_{19} -x_{10} x_{18} = 0
For the calculator:
(DynkinType =A^{2}_2+A^{2}_1; ElementsCartan =((2, 4, 4, 4, 2), (0, -2, 0, 0, 0), (0, 0, 0, 2, 2)); generators =(g_{-24}, g_{24}, g_{2}, g_{-2}, x_{3} g_{-5}+x_{4} g_{-9}+x_{5} g_{-13}, x_{10} g_{13}+x_{9} g_{9}+x_{8} g_{5}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{14}^{2}x_{20}^{2}x_{21} -2x_{13} x_{14} x_{20}^{2}x_{21} +2x_{13}^{2}x_{20}^{2}x_{21} -2x_{12} x_{13} x_{20}^{2}x_{21} +2x_{12}^{2}x_{20}^{2}x_{21} -2x_{11} x_{12} x_{20}^{2}x_{21} +2x_{11}^{2}x_{20}^{2}x_{21} -2x_{14} x_{15} x_{19} x_{20} x_{21} +2x_{13} x_{15} x_{19} x_{20} x_{21} +x_{13} x_{14} x_{19} x_{20} x_{21} -2x_{13}^{2}x_{19} x_{20} x_{21} +2x_{12} x_{13} x_{19} x_{20} x_{21} -2x_{12}^{2}x_{19} x_{20} x_{21} +2x_{11} x_{12} x_{19} x_{20} x_{21} -2x_{11}^{2}x_{19} x_{20} x_{21} +2x_{14} x_{15} x_{18} x_{20} x_{21} -4x_{13} x_{15} x_{18} x_{20} x_{21} +2x_{12} x_{15} x_{18} x_{20} x_{21} -x_{14}^{2}x_{18} x_{20} x_{21} +2x_{13} x_{14} x_{18} x_{20} x_{21} -x_{12} x_{14} x_{18} x_{20} x_{21} +2x_{13} x_{15} x_{17} x_{20} x_{21} -4x_{12} x_{15} x_{17} x_{20} x_{21} +2x_{11} x_{15} x_{17} x_{20} x_{21} -x_{13} x_{14} x_{17} x_{20} x_{21} +2x_{12} x_{14} x_{17} x_{20} x_{21} -x_{11} x_{14} x_{17} x_{20} x_{21} +2x_{12} x_{15} x_{16} x_{20} x_{21} -4x_{11} x_{15} x_{16} x_{20} x_{21} -x_{12} x_{14} x_{16} x_{20} x_{21} +2x_{11} x_{14} x_{16} x_{20} x_{21} +x_{15}^{2}x_{19}^{2}x_{21} -x_{13} x_{15} x_{19}^{2}x_{21} +3/4x_{13}^{2}x_{19}^{2}x_{21} -x_{12} x_{13} x_{19}^{2}x_{21} +x_{12}^{2}x_{19}^{2}x_{21} -x_{11} x_{12} x_{19}^{2}x_{21} +x_{11}^{2}x_{19}^{2}x_{21} -2x_{15}^{2}x_{18} x_{19} x_{21} +x_{14} x_{15} x_{18} x_{19} x_{21} +2x_{13} x_{15} x_{18} x_{19} x_{21} -x_{12} x_{15} x_{18} x_{19} x_{21} -3/2x_{13} x_{14} x_{18} x_{19} x_{21} +x_{12} x_{14} x_{18} x_{19} x_{21} +1/2x_{12} x_{13} x_{18} x_{19} x_{21} -x_{12}^{2}x_{18} x_{19} x_{21} +x_{11} x_{12} x_{18} x_{19} x_{21} -x_{11}^{2}x_{18} x_{19} x_{21} -x_{13} x_{15} x_{17} x_{19} x_{21} +2x_{12} x_{15} x_{17} x_{19} x_{21} -x_{11} x_{15} x_{17} x_{19} x_{21} +x_{13} x_{14} x_{17} x_{19} x_{21} -2x_{12} x_{14} x_{17} x_{19} x_{21} +x_{11} x_{14} x_{17} x_{19} x_{21} -1/2x_{13}^{2}x_{17} x_{19} x_{21} +x_{12} x_{13} x_{17} x_{19} x_{21} -1/2x_{11} x_{13} x_{17} x_{19} x_{21} -x_{12} x_{15} x_{16} x_{19} x_{21} +2x_{11} x_{15} x_{16} x_{19} x_{21} +x_{12} x_{14} x_{16} x_{19} x_{21} -2x_{11} x_{14} x_{16} x_{19} x_{21} -1/2x_{12} x_{13} x_{16} x_{19} x_{21} +x_{11} x_{13} x_{16} x_{19} x_{21} +2x_{15}^{2}x_{18}^{2}x_{21} -2x_{14} x_{15} x_{18}^{2}x_{21} +3/4x_{14}^{2}x_{18}^{2}x_{21} -1/2x_{12} x_{14} x_{18}^{2}x_{21} +3/4x_{12}^{2}x_{18}^{2}x_{21} -x_{11} x_{12} x_{18}^{2}x_{21} +x_{11}^{2}x_{18}^{2}x_{21} -2x_{15}^{2}x_{17} x_{18} x_{21} +2x_{14} x_{15} x_{17} x_{18} x_{21} -x_{14}^{2}x_{17} x_{18} x_{21} +1/2x_{13} x_{14} x_{17} x_{18} x_{21} +x_{12} x_{14} x_{17} x_{18} x_{21} -1/2x_{11} x_{14} x_{17} x_{18} x_{21} -3/2x_{12} x_{13} x_{17} x_{18} x_{21} +x_{11} x_{13} x_{17} x_{18} x_{21} +1/2x_{11} x_{12} x_{17} x_{18} x_{21} -x_{11}^{2}x_{17} x_{18} x_{21} -1/2x_{12} x_{14} x_{16} x_{18} x_{21} +x_{11} x_{14} x_{16} x_{18} x_{21} +x_{12} x_{13} x_{16} x_{18} x_{21} -2x_{11} x_{13} x_{16} x_{18} x_{21} -1/2x_{12}^{2}x_{16} x_{18} x_{21} +x_{11} x_{12} x_{16} x_{18} x_{21} +2x_{15}^{2}x_{17}^{2}x_{21} -2x_{14} x_{15} x_{17}^{2}x_{21} +x_{14}^{2}x_{17}^{2}x_{21} -x_{13} x_{14} x_{17}^{2}x_{21} +3/4x_{13}^{2}x_{17}^{2}x_{21} -1/2x_{11} x_{13} x_{17}^{2}x_{21} +3/4x_{11}^{2}x_{17}^{2}x_{21} -2x_{15}^{2}x_{16} x_{17} x_{21} +2x_{14} x_{15} x_{16} x_{17} x_{21} -x_{14}^{2}x_{16} x_{17} x_{21} +x_{13} x_{14} x_{16} x_{17} x_{21} -x_{13}^{2}x_{16} x_{17} x_{21} +1/2x_{12} x_{13} x_{16} x_{17} x_{21} +x_{11} x_{13} x_{16} x_{17} x_{21} -3/2x_{11} x_{12} x_{16} x_{17} x_{21} +2x_{15}^{2}x_{16}^{2}x_{21} -2x_{14} x_{15} x_{16}^{2}x_{21} +x_{14}^{2}x_{16}^{2}x_{21} -x_{13} x_{14} x_{16}^{2}x_{21} +x_{13}^{2}x_{16}^{2}x_{21} -x_{12} x_{13} x_{16}^{2}x_{21} +3/4x_{12}^{2}x_{16}^{2}x_{21} -1, x_{12} , x_{17} , x_{13} -2x_{12} +x_{11} , x_{18} -2x_{17} +x_{16} , x_{5} x_{10} +2x_{4} x_{9} +x_{3} x_{8} -2, x_{5} x_{9} +x_{4} x_{8} , x_{4} x_{10} +x_{3} x_{9} , x_{5} x_{10} +x_{4} x_{9} -1, 2x_{3} x_{15} -x_{3} x_{14} , 2x_{4} x_{15} -x_{4} x_{13} , x_{5} x_{14} -x_{5} x_{13} , 2x_{8} x_{15} -x_{8} x_{14} , 2x_{9} x_{15} -x_{9} x_{13} , x_{10} x_{14} -x_{10} x_{13} , 2x_{3} x_{20} -x_{3} x_{19} , 2x_{4} x_{20} -x_{4} x_{18} , x_{5} x_{19} -x_{5} x_{18} , 2x_{8} x_{20} -x_{8} x_{19} , 2x_{9} x_{20} -x_{9} x_{18} , x_{10} x_{19} -x_{10} x_{18} )