Processing math: 100%
Subalgebra A81E16
7 out of 119
Computations done by the calculator project.

Subalgebra type: A81 (click on type for detailed printout).
Centralizer: G12 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A22
Basis of Cartan of centralizer: 2 vectors: (0, 1, 0, 0, 0, 0), (0, 0, 1, 1, 1, 0)
Contained up to conjugation as a direct summand of: A81+A11 , A81+A31 , A81+A41 , A281+A81 , A81+A31+A11 , A12+A81 , G12+A81 .

Elements Cartan subalgebra scaled to act by two by components: A81: (4, 4, 6, 8, 6, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g7+g11+g26+g28
Positive simple generators: 2g28+2g26+2g11+2g7
Cartan symmetric matrix: (1/4)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (16)
Decomposition of ambient Lie algebra: 8V4ω18V2ω114V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+2ψ2V4ω1+2ψ1V2ψ1+2ψ2V2ω1+2ψ2V4ω12ψ1+2ψ2V2ω1+2ψ12V4ω1V4ω1+2ψ12ψ2V2ψ1+4ψ2V2ψ2V2ω12ψ1+2ψ2V2ψ12V2ω1V4ω12ψ1V4ψ12ψ2V2ω1+2ψ12ψ2V4ω12ψ2V2ψ1+2ψ22V0V2ω12ψ1V2ψ12ψ2V2ω12ψ2V4ψ1+2ψ2V2ψ1V2ψ2V2ψ14ψ2V2ψ12ψ2
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 30) ; the vectors are over the primal subalgebra.g19g15g3g5+g23g4g13+g14g2g8g9+g10h5+h4+h3h2g10+g9+g8g2g14g13+g4g23+g5+g3g15g19g6+g1g16+g12g20+g17g26+g11g28+g7g21+g18g25+g22g31+g29g30g24g27g33g32g35g36g34
weight000000000000002ω12ω12ω12ω12ω12ω12ω12ω14ω14ω14ω14ω14ω14ω14ω14ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). 2ψ12ψ22ψ14ψ22ψ22ψ14ψ1+2ψ22ψ12ψ2002ψ1+2ψ24ψ12ψ22ψ12ψ22ψ1+4ψ22ψ1+2ψ22ω12ψ22ω12ψ12ω1+2ψ12ψ22ω12ω12ω12ψ1+2ψ22ω1+2ψ12ω1+2ψ24ω12ψ24ω12ψ14ω1+2ψ12ψ24ω14ω14ω12ψ1+2ψ24ω1+2ψ14ω1+2ψ2
Isotypic module decomposition over primal subalgebra (total 28 isotypic components).
Isotypical components + highest weightV2ψ12ψ2 → (0, -2, -2)V2ψ14ψ2 → (0, 2, -4)V2ψ2 → (0, 0, -2)V2ψ1 → (0, -2, 0)V4ψ1+2ψ2 → (0, -4, 2)V2ψ12ψ2 → (0, 2, -2)V0 → (0, 0, 0)V2ψ1+2ψ2 → (0, -2, 2)V4ψ12ψ2 → (0, 4, -2)V2ψ1 → (0, 2, 0)V2ψ2 → (0, 0, 2)V2ψ1+4ψ2 → (0, -2, 4)V2ψ1+2ψ2 → (0, 2, 2)V2ω12ψ2 → (2, 0, -2)V2ω12ψ1 → (2, -2, 0)V2ω1+2ψ12ψ2 → (2, 2, -2)V2ω1 → (2, 0, 0)V2ω12ψ1+2ψ2 → (2, -2, 2)V2ω1+2ψ1 → (2, 2, 0)V2ω1+2ψ2 → (2, 0, 2)V4ω12ψ2 → (4, 0, -2)V4ω12ψ1 → (4, -2, 0)V4ω1+2ψ12ψ2 → (4, 2, -2)V4ω1 → (4, 0, 0)V4ω12ψ1+2ψ2 → (4, -2, 2)V4ω1+2ψ1 → (4, 2, 0)V4ω1+2ψ2 → (4, 0, 2)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13W14W15W16W17W18W19W20W21W22W23W24W25W26W27W28
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g19
g15
g3g5+g23
g4g13+g14
g2
g8g9+g10
Cartan of centralizer component.
h5+h4+h3
h2
g10+g9+g8
g2
g14g13+g4
g23+g5+g3
g15
g19
g6+g1
g3g5
g29g31
g16+g12
g13g14
g22g25
g20+g17
g9+g10
g18+g21
Semisimple subalgebra component.
g28g26g11g7
2h6+3h5+4h4+3h3+2h2+2h1
g7+g11+g26+g28
g26+g11
h62h52h4h3h2h1
g11g26
g21+g18
g10+g9
g17g20
g25+g22
g14+g13
g12+g16
g31+g29
g5g3
g1+g6
g30
g6g1
g3g52g23
3g293g31
6g34
g24
g16g12
2g4g13+g14
3g223g25
6g36
g27
g20g17
2g8+g9g10
3g18+3g21
6g35
g33
g26+g11
h6+2h4+h3+h2+h1
3g11+3g26
6g33
g32
g28+g7
h6+h5+2h4+h2h1
3g7+3g28
6g32
g35
g21g18
g10+g92g8
3g173g20
6g27
g36
g25g22
g14+g13+2g4
3g12+3g16
6g24
g34
g31g29
2g23+g5+g3
3g1+3g6
6g30
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above00000000000002ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
4ω1
2ω1
0
2ω1
4ω1
4ω1
2ω1
0
2ω1
4ω1
4ω1
2ω1
0
2ω1
4ω1
4ω1
2ω1
0
2ω1
4ω1
4ω1
2ω1
0
2ω1
4ω1
4ω1
2ω1
0
2ω1
4ω1
4ω1
2ω1
0
2ω1
4ω1
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer2ψ12ψ22ψ14ψ22ψ22ψ14ψ1+2ψ22ψ12ψ202ψ1+2ψ24ψ12ψ22ψ12ψ22ψ1+4ψ22ψ1+2ψ22ω12ψ2
2ψ2
2ω12ψ2
2ω12ψ1
2ψ1
2ω12ψ1
2ω1+2ψ12ψ2
2ψ12ψ2
2ω1+2ψ12ψ2
2ω1
0
2ω1
2ω1
0
2ω1
2ω12ψ1+2ψ2
2ψ1+2ψ2
2ω12ψ1+2ψ2
2ω1+2ψ1
2ψ1
2ω1+2ψ1
2ω1+2ψ2
2ψ2
2ω1+2ψ2
4ω12ψ2
2ω12ψ2
2ψ2
2ω12ψ2
4ω12ψ2
4ω12ψ1
2ω12ψ1
2ψ1
2ω12ψ1
4ω12ψ1
4ω1+2ψ12ψ2
2ω1+2ψ12ψ2
2ψ12ψ2
2ω1+2ψ12ψ2
4ω1+2ψ12ψ2
4ω1
2ω1
0
2ω1
4ω1
4ω12ψ1+2ψ2
2ω12ψ1+2ψ2
2ψ1+2ψ2
2ω12ψ1+2ψ2
4ω12ψ1+2ψ2
4ω1+2ψ1
2ω1+2ψ1
2ψ1
2ω1+2ψ1
4ω1+2ψ1
4ω1+2ψ2
2ω1+2ψ2
2ψ2
2ω1+2ψ2
4ω1+2ψ2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M2ψ12ψ2M2ψ14ψ2M2ψ2M2ψ1M4ψ1+2ψ2M2ψ12ψ2M0M2ψ1+2ψ2M4ψ12ψ2M2ψ1M2ψ2M2ψ1+4ψ2M2ψ1+2ψ2M2ω12ψ2M2ψ2M2ω12ψ2M2ω12ψ1M2ψ1M2ω12ψ1M2ω1+2ψ12ψ2M2ψ12ψ2M2ω1+2ψ12ψ2M2ω1M0M2ω1M2ω1M0M2ω1M2ω12ψ1+2ψ2M2ψ1+2ψ2M2ω12ψ1+2ψ2M2ω1+2ψ1M2ψ1M2ω1+2ψ1M2ω1+2ψ2M2ψ2M2ω1+2ψ2M4ω12ψ2M2ω12ψ2M2ψ2M2ω12ψ2M4ω12ψ2M4ω12ψ1M2ω12ψ1M2ψ1M2ω12ψ1M4ω12ψ1M4ω1+2ψ12ψ2M2ω1+2ψ12ψ2M2ψ12ψ2M2ω1+2ψ12ψ2M4ω1+2ψ12ψ2M4ω1M2ω1M0M2ω1M4ω1M4ω12ψ1+2ψ2M2ω12ψ1+2ψ2M2ψ1+2ψ2M2ω12ψ1+2ψ2M4ω12ψ1+2ψ2M4ω1+2ψ1M2ω1+2ψ1M2ψ1M2ω1+2ψ1M4ω1+2ψ1M4ω1+2ψ2M2ω1+2ψ2M2ψ2M2ω1+2ψ2M4ω1+2ψ2
Isotypic characterM2ψ12ψ2M2ψ14ψ2M2ψ2M2ψ1M4ψ1+2ψ2M2ψ12ψ22M0M2ψ1+2ψ2M4ψ12ψ2M2ψ1M2ψ2M2ψ1+4ψ2M2ψ1+2ψ2M2ω12ψ2M2ψ2M2ω12ψ2M2ω12ψ1M2ψ1M2ω12ψ1M2ω1+2ψ12ψ2M2ψ12ψ2M2ω1+2ψ12ψ2M2ω1M0M2ω1M2ω1M0M2ω1M2ω12ψ1+2ψ2M2ψ1+2ψ2M2ω12ψ1+2ψ2M2ω1+2ψ1M2ψ1M2ω1+2ψ1M2ω1+2ψ2M2ψ2M2ω1+2ψ2M4ω12ψ2M2ω12ψ2M2ψ2M2ω12ψ2M4ω12ψ2M4ω12ψ1M2ω12ψ1M2ψ1M2ω12ψ1M4ω12ψ1M4ω1+2ψ12ψ2M2ω1+2ψ12ψ2M2ψ12ψ2M2ω1+2ψ12ψ2M4ω1+2ψ12ψ22M4ω12M2ω12M02M2ω12M4ω1M4ω12ψ1+2ψ2M2ω12ψ1+2ψ2M2ψ1+2ψ2M2ω12ψ1+2ψ2M4ω12ψ1+2ψ2M4ω1+2ψ1M2ω1+2ψ1M2ψ1M2ω1+2ψ1M4ω1+2ψ1M4ω1+2ψ2M2ω1+2ψ2M2ψ2M2ω1+2ψ2M4ω1+2ψ2

Semisimple subalgebra: W_{17}
Centralizer extension: W_{1}+W_{2}+W_{3}+W_{4}+W_{5}+W_{6}+W_{7}+W_{8}+W_{9}+W_{10}+W_{11}+W_{12}+W_{13}

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, 1.00, 0.00)
0: (1.00, 0.00, 0.00): (600.00, 300.00)
1: (0.00, 1.00, 0.00): (200.00, 316.67)
2: (0.00, 0.00, 1.00): (200.00, 308.33)



Made total 22819045 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.
1*2 (unknown) gens:
(
x_{1} g_{-1}+x_{2} g_{-6}+x_{3} g_{-7}+x_{4} g_{-11}+x_{5} g_{-12}+x_{6} g_{-16}+x_{7} g_{-17}+x_{8} g_{-18} \\ +x_{9} g_{-20}+x_{10} g_{-21}+x_{11} g_{-22}+x_{12} g_{-25}+x_{13} g_{-26}+x_{14} g_{-28}+x_{15} g_{-29} \\ +x_{16} g_{-31}, x_{32} g_{31}+x_{31} g_{29}+x_{30} g_{28}+x_{29} g_{26}+x_{28} g_{25}+x_{27} g_{22}+x_{26} g_{21}+x_{25} g_{20} \\ +x_{24} g_{18}+x_{23} g_{17}+x_{22} g_{16}+x_{21} g_{12}+x_{20} g_{11}+x_{19} g_{7}+x_{18} g_{6}+x_{17} g_{1})
h: (4, 4, 6, 8, 6, 4), e = combination of g_{1} g_{6} g_{7} g_{11} g_{12} g_{16} g_{17} g_{18} g_{20} g_{21} g_{22} g_{25} g_{26} g_{28} g_{29} g_{31} , f= combination of g_{-1} g_{-6} g_{-7} g_{-11} g_{-12} g_{-16} g_{-17} g_{-18} g_{-20} g_{-21} g_{-22} g_{-25} g_{-26} g_{-28} g_{-29} g_{-31} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: 8V_{4\omega_{1}}+16V_{2\omega_{1}}+30V_{0}+16V_{-2\omega_{1}}+8V_{-4\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{15} x_{31} +x_{13} x_{29} +x_{11} x_{27} +x_{8} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{3} x_{19} +x_{1} x_{17}
-4= 0
x_{15} x_{29} +x_{12} x_{25} +x_{10} x_{22} -x_{3} x_{17} = 0
x_{15} x_{27} -x_{14} x_{25} +x_{10} x_{20} -x_{5} x_{17} = 0
x_{15} x_{24} -x_{14} x_{22} -x_{12} x_{20} +x_{7} x_{17} = 0
x_{16} x_{25} -x_{15} x_{23} +x_{10} x_{18} -x_{8} x_{17} = 0
x_{16} x_{22} -x_{15} x_{21} -x_{12} x_{18} +x_{11} x_{17} = 0
x_{16} x_{20} +x_{15} x_{19} +x_{14} x_{18} -x_{13} x_{17} = 0
x_{16} x_{32} +x_{14} x_{30} +x_{12} x_{28} +x_{10} x_{26} +x_{9} x_{25} +x_{6} x_{22} +x_{4} x_{20}
+x_{2} x_{18} -4= 0
x_{16} x_{30} -x_{11} x_{23} -x_{8} x_{21} +x_{4} x_{18} = 0
x_{16} x_{28} +x_{13} x_{23} +x_{8} x_{19} -x_{6} x_{18} = 0
x_{16} x_{26} +x_{13} x_{21} -x_{11} x_{19} +x_{9} x_{18} = 0
x_{13} x_{31} +x_{9} x_{28} +x_{6} x_{26} -x_{1} x_{19} = 0
x_{16} x_{32} +2x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26}
+x_{8} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{3} x_{19} -6= 0
x_{14} x_{28} +x_{13} x_{27} +x_{6} x_{20} -x_{5} x_{19} = 0
x_{14} x_{26} +x_{13} x_{24} -x_{9} x_{20} +x_{7} x_{19} = 0
x_{14} x_{32} -x_{7} x_{27} -x_{5} x_{24} +x_{2} x_{20} = 0
2x_{16} x_{32} +x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26}
+x_{9} x_{25} +x_{8} x_{24} +x_{6} x_{22} +x_{4} x_{20} -6= 0
x_{11} x_{31} -x_{9} x_{30} +x_{4} x_{26} -x_{1} x_{21} = 0
x_{12} x_{30} +x_{11} x_{29} +x_{4} x_{22} -x_{3} x_{21} = 0
2x_{16} x_{32} +2x_{15} x_{31} +2x_{14} x_{30} +2x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26}
+x_{9} x_{25} +x_{8} x_{24} +x_{7} x_{23} +x_{6} x_{22} +x_{5} x_{21} -8= 0
x_{12} x_{26} +x_{11} x_{24} +x_{9} x_{22} +x_{7} x_{21} = 0
x_{12} x_{32} +x_{7} x_{29} +x_{3} x_{24} -x_{2} x_{22} = 0
x_{8} x_{31} -x_{6} x_{30} -x_{4} x_{28} +x_{1} x_{23} = 0
x_{10} x_{30} +x_{8} x_{29} -x_{4} x_{25} +x_{3} x_{23} = 0
x_{10} x_{28} +x_{8} x_{27} +x_{6} x_{25} +x_{5} x_{23} = 0
x_{16} x_{32} +x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{9} x_{25}
+x_{7} x_{23} -4= 0
x_{9} x_{32} -x_{7} x_{31} +x_{2} x_{26} -x_{1} x_{24} = 0
x_{10} x_{32} +x_{5} x_{29} -x_{3} x_{27} +x_{2} x_{25} = 0
x_{6} x_{32} -x_{5} x_{31} -x_{2} x_{28} +x_{1} x_{27} = 0
x_{4} x_{32} +x_{3} x_{31} +x_{2} x_{30} -x_{1} x_{29} = 0
The above system after transformation.
x_{15} x_{31} +x_{13} x_{29} +x_{11} x_{27} +x_{8} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{3} x_{19} +x_{1} x_{17}
-4= 0
x_{15} x_{29} +x_{12} x_{25} +x_{10} x_{22} -x_{3} x_{17} = 0
x_{15} x_{27} -x_{14} x_{25} +x_{10} x_{20} -x_{5} x_{17} = 0
x_{15} x_{24} -x_{14} x_{22} -x_{12} x_{20} +x_{7} x_{17} = 0
x_{16} x_{25} -x_{15} x_{23} +x_{10} x_{18} -x_{8} x_{17} = 0
x_{16} x_{22} -x_{15} x_{21} -x_{12} x_{18} +x_{11} x_{17} = 0
x_{16} x_{20} +x_{15} x_{19} +x_{14} x_{18} -x_{13} x_{17} = 0
x_{16} x_{32} +x_{14} x_{30} +x_{12} x_{28} +x_{10} x_{26} +x_{9} x_{25} +x_{6} x_{22} +x_{4} x_{20}
+x_{2} x_{18} -4= 0
x_{16} x_{30} -x_{11} x_{23} -x_{8} x_{21} +x_{4} x_{18} = 0
x_{16} x_{28} +x_{13} x_{23} +x_{8} x_{19} -x_{6} x_{18} = 0
x_{16} x_{26} +x_{13} x_{21} -x_{11} x_{19} +x_{9} x_{18} = 0
x_{13} x_{31} +x_{9} x_{28} +x_{6} x_{26} -x_{1} x_{19} = 0
x_{16} x_{32} +2x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26}
+x_{8} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{3} x_{19} -6= 0
x_{14} x_{28} +x_{13} x_{27} +x_{6} x_{20} -x_{5} x_{19} = 0
x_{14} x_{26} +x_{13} x_{24} -x_{9} x_{20} +x_{7} x_{19} = 0
x_{14} x_{32} -x_{7} x_{27} -x_{5} x_{24} +x_{2} x_{20} = 0
2x_{16} x_{32} +x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26}
+x_{9} x_{25} +x_{8} x_{24} +x_{6} x_{22} +x_{4} x_{20} -6= 0
x_{11} x_{31} -x_{9} x_{30} +x_{4} x_{26} -x_{1} x_{21} = 0
x_{12} x_{30} +x_{11} x_{29} +x_{4} x_{22} -x_{3} x_{21} = 0
2x_{16} x_{32} +2x_{15} x_{31} +2x_{14} x_{30} +2x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26}
+x_{9} x_{25} +x_{8} x_{24} +x_{7} x_{23} +x_{6} x_{22} +x_{5} x_{21} -8= 0
x_{12} x_{26} +x_{11} x_{24} +x_{9} x_{22} +x_{7} x_{21} = 0
x_{12} x_{32} +x_{7} x_{29} +x_{3} x_{24} -x_{2} x_{22} = 0
x_{8} x_{31} -x_{6} x_{30} -x_{4} x_{28} +x_{1} x_{23} = 0
x_{10} x_{30} +x_{8} x_{29} -x_{4} x_{25} +x_{3} x_{23} = 0
x_{10} x_{28} +x_{8} x_{27} +x_{6} x_{25} +x_{5} x_{23} = 0
x_{16} x_{32} +x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{9} x_{25}
+x_{7} x_{23} -4= 0
x_{9} x_{32} -x_{7} x_{31} +x_{2} x_{26} -x_{1} x_{24} = 0
x_{10} x_{32} +x_{5} x_{29} -x_{3} x_{27} +x_{2} x_{25} = 0
x_{6} x_{32} -x_{5} x_{31} -x_{2} x_{28} +x_{1} x_{27} = 0
x_{4} x_{32} +x_{3} x_{31} +x_{2} x_{30} -x_{1} x_{29} = 0
For the calculator:
(DynkinType =A^{8}_1; ElementsCartan =((4, 4, 6, 8, 6, 4)); generators =(x_{1} g_{-1}+x_{2} g_{-6}+x_{3} g_{-7}+x_{4} g_{-11}+x_{5} g_{-12}+x_{6} g_{-16}+x_{7} g_{-17}+x_{8} g_{-18}+x_{9} g_{-20}+x_{10} g_{-21}+x_{11} g_{-22}+x_{12} g_{-25}+x_{13} g_{-26}+x_{14} g_{-28}+x_{15} g_{-29}+x_{16} g_{-31}, x_{32} g_{31}+x_{31} g_{29}+x_{30} g_{28}+x_{29} g_{26}+x_{28} g_{25}+x_{27} g_{22}+x_{26} g_{21}+x_{25} g_{20}+x_{24} g_{18}+x_{23} g_{17}+x_{22} g_{16}+x_{21} g_{12}+x_{20} g_{11}+x_{19} g_{7}+x_{18} g_{6}+x_{17} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{15} x_{31} +x_{13} x_{29} +x_{11} x_{27} +x_{8} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{3} x_{19} +x_{1} x_{17} -4, x_{15} x_{29} +x_{12} x_{25} +x_{10} x_{22} -x_{3} x_{17} , x_{15} x_{27} -x_{14} x_{25} +x_{10} x_{20} -x_{5} x_{17} , x_{15} x_{24} -x_{14} x_{22} -x_{12} x_{20} +x_{7} x_{17} , x_{16} x_{25} -x_{15} x_{23} +x_{10} x_{18} -x_{8} x_{17} , x_{16} x_{22} -x_{15} x_{21} -x_{12} x_{18} +x_{11} x_{17} , x_{16} x_{20} +x_{15} x_{19} +x_{14} x_{18} -x_{13} x_{17} , x_{16} x_{32} +x_{14} x_{30} +x_{12} x_{28} +x_{10} x_{26} +x_{9} x_{25} +x_{6} x_{22} +x_{4} x_{20} +x_{2} x_{18} -4, x_{16} x_{30} -x_{11} x_{23} -x_{8} x_{21} +x_{4} x_{18} , x_{16} x_{28} +x_{13} x_{23} +x_{8} x_{19} -x_{6} x_{18} , x_{16} x_{26} +x_{13} x_{21} -x_{11} x_{19} +x_{9} x_{18} , x_{13} x_{31} +x_{9} x_{28} +x_{6} x_{26} -x_{1} x_{19} , x_{16} x_{32} +2x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26} +x_{8} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{3} x_{19} -6, x_{14} x_{28} +x_{13} x_{27} +x_{6} x_{20} -x_{5} x_{19} , x_{14} x_{26} +x_{13} x_{24} -x_{9} x_{20} +x_{7} x_{19} , x_{14} x_{32} -x_{7} x_{27} -x_{5} x_{24} +x_{2} x_{20} , 2x_{16} x_{32} +x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26} +x_{9} x_{25} +x_{8} x_{24} +x_{6} x_{22} +x_{4} x_{20} -6, x_{11} x_{31} -x_{9} x_{30} +x_{4} x_{26} -x_{1} x_{21} , x_{12} x_{30} +x_{11} x_{29} +x_{4} x_{22} -x_{3} x_{21} , 2x_{16} x_{32} +2x_{15} x_{31} +2x_{14} x_{30} +2x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{10} x_{26} +x_{9} x_{25} +x_{8} x_{24} +x_{7} x_{23} +x_{6} x_{22} +x_{5} x_{21} -8, x_{12} x_{26} +x_{11} x_{24} +x_{9} x_{22} +x_{7} x_{21} , x_{12} x_{32} +x_{7} x_{29} +x_{3} x_{24} -x_{2} x_{22} , x_{8} x_{31} -x_{6} x_{30} -x_{4} x_{28} +x_{1} x_{23} , x_{10} x_{30} +x_{8} x_{29} -x_{4} x_{25} +x_{3} x_{23} , x_{10} x_{28} +x_{8} x_{27} +x_{6} x_{25} +x_{5} x_{23} , x_{16} x_{32} +x_{15} x_{31} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{11} x_{27} +x_{9} x_{25} +x_{7} x_{23} -4, x_{9} x_{32} -x_{7} x_{31} +x_{2} x_{26} -x_{1} x_{24} , x_{10} x_{32} +x_{5} x_{29} -x_{3} x_{27} +x_{2} x_{25} , x_{6} x_{32} -x_{5} x_{31} -x_{2} x_{28} +x_{1} x_{27} , x_{4} x_{32} +x_{3} x_{31} +x_{2} x_{30} -x_{1} x_{29} )