Subalgebra A131C15
12 out of 119
Computations done by the calculator project.

Subalgebra type: A131 (click on type for detailed printout).
Centralizer: A81 .
The semisimple part of the centralizer of the semisimple part of my centralizer: B12+A31
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)
Contained up to conjugation as a direct summand of: A131+A81 .

Elements Cartan subalgebra scaled to act by two by components: A131: (6, 8, 10, 12, 7): 26
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g1+g9+g19+g23
Positive simple generators: 4g23+g19+g9+3g1
Cartan symmetric matrix: (2/13)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (26)
Decomposition of ambient Lie algebra: V6ω13V4ω110V2ω13V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V6ω1V4ω1+ψV2ω1+2ψV4ω12V2ω1+ψV4ω1ψ4V2ω1Vψ2V2ω1ψV0V2ω12ψVψ
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 17) ; the vectors are over the primal subalgebra.g3+g8h4g8+g3g5g12g15+3g10g9g19g21+3g6g23+3/4g1g16g18+3g14g13g17g22g20g25
weight0002ω12ω12ω12ω12ω12ω12ω12ω12ω12ω14ω14ω14ω16ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). ψ0ψ2ω12ψ2ω1ψ2ω1ψ2ω12ω12ω12ω12ω1+ψ2ω1+ψ2ω1+2ψ4ω1ψ4ω14ω1+ψ6ω1
Isotypic module decomposition over primal subalgebra (total 13 isotypic components).
Isotypical components + highest weightVψ → (0, -1)V0 → (0, 0)Vψ → (0, 1)V2ω12ψ → (2, -2)V2ω1ψ → (2, -1)V2ω1 → (2, 0)V2ω1+ψ → (2, 1)V2ω1+2ψ → (2, 2)V4ω1ψ → (4, -1)V4ω1 → (4, 0)V4ω1+ψ → (4, 1)V6ω1 → (6, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g3+g8
Cartan of centralizer component.
h4
g8+g3
g5
g4
2g13
g12
g3g8
2g16
g15+3g10
2g7g11
g14+g18
Semisimple subalgebra component.
4/3g231/3g191/3g9g1
7/3h5+4h4+10/3h3+8/3h2+2h1
2/3g1+2/3g9+2/3g19+2/3g23
g21+3g6
2g2g2
g6+g21
g9
2h52h4
2g9
g19
h52h42h3
2g19
g16
g8g3
2g12
g18+3g14
2g11g7
g10+g15
g13
g4
2g5
g17
g15g10
2g7g11
g143g18
4g20
g22
g21g6
2g2g2
g63g21
4g22
g20
g18g14
2g11g7
g103g15
4g17
g25
g24
2g23g1
2h54h44h34h2+2h1
4g16g23
10g24
20g25
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0002ω1
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
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizerψ0ψ2ω12ψ
2ψ
2ω12ψ
2ω1ψ
ψ
2ω1ψ
2ω1
0
2ω1
2ω1
0
2ω1
2ω1+ψ
ψ
2ω1+ψ
2ω1+2ψ
2ψ
2ω1+2ψ
4ω1ψ
2ω1ψ
ψ
2ω1ψ
4ω1ψ
4ω1
2ω1
0
2ω1
4ω1
4ω1+ψ
2ω1+ψ
ψ
2ω1+ψ
4ω1+ψ
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.MψM0MψM2ω12ψM2ψM2ω12ψM2ω1ψMψM2ω1ψM2ω1M0M2ω1M2ω1M0M2ω1M2ω1+ψMψM2ω1+ψM2ω1+2ψM2ψM2ω1+2ψM4ω1ψM2ω1ψMψM2ω1ψM4ω1ψM4ω1M2ω1M0M2ω1M4ω1M4ω1+ψM2ω1+ψMψM2ω1+ψM4ω1+ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1
Isotypic characterMψM0MψM2ω12ψM2ψM2ω12ψ2M2ω1ψ2Mψ2M2ω1ψM2ω1M0M2ω13M2ω13M03M2ω12M2ω1+ψ2Mψ2M2ω1+ψM2ω1+2ψM2ψM2ω1+2ψM4ω1ψM2ω1ψMψM2ω1ψM4ω1ψM4ω1M2ω1M0M2ω1M4ω1M4ω1+ψM2ω1+ψMψM2ω1+ψM4ω1+ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1

Semisimple subalgebra: W_{6}
Centralizer extension: W_{1}+W_{2}+W_{3}

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, 1.00)
0: (1.00, 0.00): (850.00, 300.00)
1: (0.00, 1.00): (200.00, 325.00)




Made total 54301561 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_{-5}+x_{3} g_{-6}+x_{4} g_{-9}+x_{5} g_{-10}+x_{6} g_{-12}+x_{7} g_{-13}+x_{8} g_{-14} \\ +x_{9} g_{-15}+x_{10} g_{-16}+x_{11} g_{-18}+x_{12} g_{-19}+x_{13} g_{-21}+x_{14} g_{-23}, x_{28} g_{23}+x_{27} g_{21}+x_{26} g_{19}+x_{25} g_{18}+x_{24} g_{16}+x_{23} g_{15}+x_{22} g_{14}+x_{21} g_{13} \\ +x_{20} g_{12}+x_{19} g_{10}+x_{18} g_{9}+x_{17} g_{6}+x_{16} g_{5}+x_{15} g_{1})

Unknown splitting cartan of centralizer.
x_{33} h_{5}+x_{32} h_{4}+x_{31} h_{3}+x_{30} h_{2}+x_{29} h_{1}
h: (6, 8, 10, 12, 7), e = combination of g_{1} g_{5} g_{6} g_{9} g_{10} g_{12} g_{13} g_{14} g_{15} g_{16} g_{18} g_{19} g_{21} g_{23} , f= combination of g_{-1} g_{-5} g_{-6} g_{-9} g_{-10} g_{-12} g_{-13} g_{-14} g_{-15} g_{-16} g_{-18} g_{-19} g_{-21} g_{-23} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{6\omega_{1}}+4V_{4\omega_{1}}+14V_{2\omega_{1}}+17V_{0}+14V_{-2\omega_{1}}+4V_{-4\omega_{1}}+V_{-6\omega_{1}}
A necessary system to realize the candidate subalgebra.
2x_{33}^{2}x_{34} -2x_{32} x_{33} x_{34} +x_{32}^{2}x_{34} -x_{31} x_{32} x_{34} +x_{31}^{2}x_{34} -x_{30} x_{31} x_{34}
+x_{30}^{2}x_{34} -x_{29} x_{30} x_{34} +x_{29}^{2}x_{34} -1= 0
x_{8} x_{22} +x_{5} x_{19} +x_{3} x_{17} +x_{1} x_{15} -3= 0
x_{14} x_{27} +x_{13} x_{26} +x_{11} x_{24} +x_{9} x_{20} -x_{3} x_{15} = 0
x_{14} x_{25} +x_{13} x_{24} +x_{11} x_{21} +x_{9} x_{18} -x_{5} x_{15} = 0
x_{14} x_{23} +x_{13} x_{20} +x_{11} x_{18} +x_{9} x_{16} -x_{8} x_{15} = 0
x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +2x_{11} x_{25} +2x_{10} x_{24} +2x_{9} x_{23} +x_{7} x_{21}
+2x_{6} x_{20} +2x_{4} x_{18} +x_{2} x_{16} -7= 0
x_{11} x_{23} +x_{10} x_{20} -x_{8} x_{19} +x_{7} x_{18} +x_{4} x_{16} = 0
x_{13} x_{23} +x_{12} x_{20} +x_{10} x_{18} -x_{8} x_{17} +x_{6} x_{16} = 0
x_{13} x_{28} +x_{12} x_{27} +x_{10} x_{25} +x_{6} x_{23} -x_{1} x_{17} = 0
x_{14} x_{28} +x_{13} x_{27} +x_{11} x_{25} +x_{9} x_{23} +x_{8} x_{22} +x_{5} x_{19} +x_{3} x_{17} -4= 0
x_{13} x_{25} +x_{12} x_{24} +x_{10} x_{21} +x_{6} x_{18} -x_{5} x_{17} = 0
x_{9} x_{25} +x_{6} x_{24} -x_{5} x_{22} +x_{4} x_{21} +x_{2} x_{18} = 0
x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +2x_{11} x_{25} +2x_{10} x_{24} +x_{9} x_{23} +x_{8} x_{22}
+x_{7} x_{21} +x_{6} x_{20} +x_{4} x_{18} -6= 0
x_{11} x_{28} +x_{10} x_{27} +x_{7} x_{25} +x_{4} x_{23} -x_{1} x_{19} = 0
x_{11} x_{27} +x_{10} x_{26} +x_{7} x_{24} +x_{4} x_{20} -x_{3} x_{19} = 0
x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +x_{11} x_{25} +x_{10} x_{24} +x_{9} x_{23} +x_{8} x_{22}
+x_{6} x_{20} +x_{5} x_{19} -5= 0
x_{9} x_{27} +x_{6} x_{26} +x_{4} x_{24} -x_{3} x_{22} +x_{2} x_{20} = 0
x_{9} x_{28} +x_{6} x_{27} +x_{4} x_{25} +x_{2} x_{23} -x_{1} x_{22} = 0
x_{1} x_{30} -2x_{1} x_{29} = 0
2x_{2} x_{33} -x_{2} x_{32} = 0
x_{3} x_{31} -x_{3} x_{30} -x_{3} x_{29} = 0
2x_{4} x_{33} -x_{4} x_{31} = 0
x_{5} x_{32} -x_{5} x_{31} -x_{5} x_{29} = 0
2x_{6} x_{33} -x_{6} x_{32} +x_{6} x_{31} -x_{6} x_{30} = 0
x_{7} x_{32} -x_{7} x_{31} = 0
2x_{8} x_{33} -x_{8} x_{32} -x_{8} x_{29} = 0
2x_{9} x_{33} -x_{9} x_{32} +x_{9} x_{30} -x_{9} x_{29} = 0
x_{10} x_{32} -x_{10} x_{30} = 0
x_{11} x_{32} -x_{11} x_{31} +x_{11} x_{30} -x_{11} x_{29} = 0
x_{12} x_{31} -x_{12} x_{30} = 0
x_{13} x_{31} -x_{13} x_{29} = 0
x_{14} x_{30} -x_{14} x_{29} = 0
x_{15} x_{30} -2x_{15} x_{29} = 0
2x_{16} x_{33} -x_{16} x_{32} = 0
x_{17} x_{31} -x_{17} x_{30} -x_{17} x_{29} = 0
2x_{18} x_{33} -x_{18} x_{31} = 0
x_{19} x_{32} -x_{19} x_{31} -x_{19} x_{29} = 0
2x_{20} x_{33} -x_{20} x_{32} +x_{20} x_{31} -x_{20} x_{30} = 0
x_{21} x_{32} -x_{21} x_{31} = 0
2x_{22} x_{33} -x_{22} x_{32} -x_{22} x_{29} = 0
2x_{23} x_{33} -x_{23} x_{32} +x_{23} x_{30} -x_{23} x_{29} = 0
x_{24} x_{32} -x_{24} x_{30} = 0
x_{25} x_{32} -x_{25} x_{31} +x_{25} x_{30} -x_{25} x_{29} = 0
x_{26} x_{31} -x_{26} x_{30} = 0
x_{27} x_{31} -x_{27} x_{29} = 0
x_{28} x_{30} -x_{28} x_{29} = 0
The above system after transformation.
2x_{33}^{2}x_{34} -2x_{32} x_{33} x_{34} +x_{32}^{2}x_{34} -x_{31} x_{32} x_{34} +x_{31}^{2}x_{34} -x_{30} x_{31} x_{34}
+x_{30}^{2}x_{34} -x_{29} x_{30} x_{34} +x_{29}^{2}x_{34} -1= 0
x_{8} x_{22} +x_{5} x_{19} +x_{3} x_{17} +x_{1} x_{15} -3= 0
x_{14} x_{27} +x_{13} x_{26} +x_{11} x_{24} +x_{9} x_{20} -x_{3} x_{15} = 0
x_{14} x_{25} +x_{13} x_{24} +x_{11} x_{21} +x_{9} x_{18} -x_{5} x_{15} = 0
x_{14} x_{23} +x_{13} x_{20} +x_{11} x_{18} +x_{9} x_{16} -x_{8} x_{15} = 0
x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +2x_{11} x_{25} +2x_{10} x_{24} +2x_{9} x_{23} +x_{7} x_{21}
+2x_{6} x_{20} +2x_{4} x_{18} +x_{2} x_{16} -7= 0
x_{11} x_{23} +x_{10} x_{20} -x_{8} x_{19} +x_{7} x_{18} +x_{4} x_{16} = 0
x_{13} x_{23} +x_{12} x_{20} +x_{10} x_{18} -x_{8} x_{17} +x_{6} x_{16} = 0
x_{13} x_{28} +x_{12} x_{27} +x_{10} x_{25} +x_{6} x_{23} -x_{1} x_{17} = 0
x_{14} x_{28} +x_{13} x_{27} +x_{11} x_{25} +x_{9} x_{23} +x_{8} x_{22} +x_{5} x_{19} +x_{3} x_{17} -4= 0
x_{13} x_{25} +x_{12} x_{24} +x_{10} x_{21} +x_{6} x_{18} -x_{5} x_{17} = 0
x_{9} x_{25} +x_{6} x_{24} -x_{5} x_{22} +x_{4} x_{21} +x_{2} x_{18} = 0
x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +2x_{11} x_{25} +2x_{10} x_{24} +x_{9} x_{23} +x_{8} x_{22}
+x_{7} x_{21} +x_{6} x_{20} +x_{4} x_{18} -6= 0
x_{11} x_{28} +x_{10} x_{27} +x_{7} x_{25} +x_{4} x_{23} -x_{1} x_{19} = 0
x_{11} x_{27} +x_{10} x_{26} +x_{7} x_{24} +x_{4} x_{20} -x_{3} x_{19} = 0
x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +x_{11} x_{25} +x_{10} x_{24} +x_{9} x_{23} +x_{8} x_{22}
+x_{6} x_{20} +x_{5} x_{19} -5= 0
x_{9} x_{27} +x_{6} x_{26} +x_{4} x_{24} -x_{3} x_{22} +x_{2} x_{20} = 0
x_{9} x_{28} +x_{6} x_{27} +x_{4} x_{25} +x_{2} x_{23} -x_{1} x_{22} = 0
x_{1} x_{30} -2x_{1} x_{29} = 0
2x_{2} x_{33} -x_{2} x_{32} = 0
x_{3} x_{31} -x_{3} x_{30} -x_{3} x_{29} = 0
2x_{4} x_{33} -x_{4} x_{31} = 0
x_{5} x_{32} -x_{5} x_{31} -x_{5} x_{29} = 0
2x_{6} x_{33} -x_{6} x_{32} +x_{6} x_{31} -x_{6} x_{30} = 0
x_{7} x_{32} -x_{7} x_{31} = 0
2x_{8} x_{33} -x_{8} x_{32} -x_{8} x_{29} = 0
2x_{9} x_{33} -x_{9} x_{32} +x_{9} x_{30} -x_{9} x_{29} = 0
x_{10} x_{32} -x_{10} x_{30} = 0
x_{11} x_{32} -x_{11} x_{31} +x_{11} x_{30} -x_{11} x_{29} = 0
x_{12} x_{31} -x_{12} x_{30} = 0
x_{13} x_{31} -x_{13} x_{29} = 0
x_{14} x_{30} -x_{14} x_{29} = 0
x_{15} x_{30} -2x_{15} x_{29} = 0
2x_{16} x_{33} -x_{16} x_{32} = 0
x_{17} x_{31} -x_{17} x_{30} -x_{17} x_{29} = 0
2x_{18} x_{33} -x_{18} x_{31} = 0
x_{19} x_{32} -x_{19} x_{31} -x_{19} x_{29} = 0
2x_{20} x_{33} -x_{20} x_{32} +x_{20} x_{31} -x_{20} x_{30} = 0
x_{21} x_{32} -x_{21} x_{31} = 0
2x_{22} x_{33} -x_{22} x_{32} -x_{22} x_{29} = 0
2x_{23} x_{33} -x_{23} x_{32} +x_{23} x_{30} -x_{23} x_{29} = 0
x_{24} x_{32} -x_{24} x_{30} = 0
x_{25} x_{32} -x_{25} x_{31} +x_{25} x_{30} -x_{25} x_{29} = 0
x_{26} x_{31} -x_{26} x_{30} = 0
x_{27} x_{31} -x_{27} x_{29} = 0
x_{28} x_{30} -x_{28} x_{29} = 0
For the calculator:
(DynkinType =A^{13}_1; ElementsCartan =((6, 8, 10, 12, 7)); generators =(x_{1} g_{-1}+x_{2} g_{-5}+x_{3} g_{-6}+x_{4} g_{-9}+x_{5} g_{-10}+x_{6} g_{-12}+x_{7} g_{-13}+x_{8} g_{-14}+x_{9} g_{-15}+x_{10} g_{-16}+x_{11} g_{-18}+x_{12} g_{-19}+x_{13} g_{-21}+x_{14} g_{-23}, x_{28} g_{23}+x_{27} g_{21}+x_{26} g_{19}+x_{25} g_{18}+x_{24} g_{16}+x_{23} g_{15}+x_{22} g_{14}+x_{21} g_{13}+x_{20} g_{12}+x_{19} g_{10}+x_{18} g_{9}+x_{17} g_{6}+x_{16} g_{5}+x_{15} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( 2x_{33}^{2}x_{34} -2x_{32} x_{33} x_{34} +x_{32}^{2}x_{34} -x_{31} x_{32} x_{34} +x_{31}^{2}x_{34} -x_{30} x_{31} x_{34} +x_{30}^{2}x_{34} -x_{29} x_{30} x_{34} +x_{29}^{2}x_{34} -1, x_{8} x_{22} +x_{5} x_{19} +x_{3} x_{17} +x_{1} x_{15} -3, x_{14} x_{27} +x_{13} x_{26} +x_{11} x_{24} +x_{9} x_{20} -x_{3} x_{15} , x_{14} x_{25} +x_{13} x_{24} +x_{11} x_{21} +x_{9} x_{18} -x_{5} x_{15} , x_{14} x_{23} +x_{13} x_{20} +x_{11} x_{18} +x_{9} x_{16} -x_{8} x_{15} , x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +2x_{11} x_{25} +2x_{10} x_{24} +2x_{9} x_{23} +x_{7} x_{21} +2x_{6} x_{20} +2x_{4} x_{18} +x_{2} x_{16} -7, x_{11} x_{23} +x_{10} x_{20} -x_{8} x_{19} +x_{7} x_{18} +x_{4} x_{16} , x_{13} x_{23} +x_{12} x_{20} +x_{10} x_{18} -x_{8} x_{17} +x_{6} x_{16} , x_{13} x_{28} +x_{12} x_{27} +x_{10} x_{25} +x_{6} x_{23} -x_{1} x_{17} , x_{14} x_{28} +x_{13} x_{27} +x_{11} x_{25} +x_{9} x_{23} +x_{8} x_{22} +x_{5} x_{19} +x_{3} x_{17} -4, x_{13} x_{25} +x_{12} x_{24} +x_{10} x_{21} +x_{6} x_{18} -x_{5} x_{17} , x_{9} x_{25} +x_{6} x_{24} -x_{5} x_{22} +x_{4} x_{21} +x_{2} x_{18} , x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +2x_{11} x_{25} +2x_{10} x_{24} +x_{9} x_{23} +x_{8} x_{22} +x_{7} x_{21} +x_{6} x_{20} +x_{4} x_{18} -6, x_{11} x_{28} +x_{10} x_{27} +x_{7} x_{25} +x_{4} x_{23} -x_{1} x_{19} , x_{11} x_{27} +x_{10} x_{26} +x_{7} x_{24} +x_{4} x_{20} -x_{3} x_{19} , x_{14} x_{28} +2x_{13} x_{27} +x_{12} x_{26} +x_{11} x_{25} +x_{10} x_{24} +x_{9} x_{23} +x_{8} x_{22} +x_{6} x_{20} +x_{5} x_{19} -5, x_{9} x_{27} +x_{6} x_{26} +x_{4} x_{24} -x_{3} x_{22} +x_{2} x_{20} , x_{9} x_{28} +x_{6} x_{27} +x_{4} x_{25} +x_{2} x_{23} -x_{1} x_{22} , x_{1} x_{30} -2x_{1} x_{29} , 2x_{2} x_{33} -x_{2} x_{32} , x_{3} x_{31} -x_{3} x_{30} -x_{3} x_{29} , 2x_{4} x_{33} -x_{4} x_{31} , x_{5} x_{32} -x_{5} x_{31} -x_{5} x_{29} , 2x_{6} x_{33} -x_{6} x_{32} +x_{6} x_{31} -x_{6} x_{30} , x_{7} x_{32} -x_{7} x_{31} , 2x_{8} x_{33} -x_{8} x_{32} -x_{8} x_{29} , 2x_{9} x_{33} -x_{9} x_{32} +x_{9} x_{30} -x_{9} x_{29} , x_{10} x_{32} -x_{10} x_{30} , x_{11} x_{32} -x_{11} x_{31} +x_{11} x_{30} -x_{11} x_{29} , x_{12} x_{31} -x_{12} x_{30} , x_{13} x_{31} -x_{13} x_{29} , x_{14} x_{30} -x_{14} x_{29} , x_{15} x_{30} -2x_{15} x_{29} , 2x_{16} x_{33} -x_{16} x_{32} , x_{17} x_{31} -x_{17} x_{30} -x_{17} x_{29} , 2x_{18} x_{33} -x_{18} x_{31} , x_{19} x_{32} -x_{19} x_{31} -x_{19} x_{29} , 2x_{20} x_{33} -x_{20} x_{32} +x_{20} x_{31} -x_{20} x_{30} , x_{21} x_{32} -x_{21} x_{31} , 2x_{22} x_{33} -x_{22} x_{32} -x_{22} x_{29} , 2x_{23} x_{33} -x_{23} x_{32} +x_{23} x_{30} -x_{23} x_{29} , x_{24} x_{32} -x_{24} x_{30} , x_{25} x_{32} -x_{25} x_{31} +x_{25} x_{30} -x_{25} x_{29} , x_{26} x_{31} -x_{26} x_{30} , x_{27} x_{31} -x_{27} x_{29} , x_{28} x_{30} -x_{28} x_{29} )