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Subalgebra A31 ↪ E16
3 out of 119
Computations done by the calculator project.

Subalgebra type: A31 (click on type for detailed printout).
Centralizer: A22+A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A31
Basis of Cartan of centralizer: 3 vectors: (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A31+A11 , A31+A21 , 2A31 , A81+A31 , A91+A31 , A31+A21+A11 , A81+A31+A11 , A22+A31 , A22+A31+A11 .

Elements Cartan subalgebra scaled to act by two by components: A31: (2, 3, 4, 6, 4, 2): 6
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−23+g−30+g−34
Positive simple generators: g34+g30+g23
Cartan symmetric matrix: (2/3)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6)
Decomposition of ambient Lie algebra: 2V3ω1⊕9V2ω1⊕16Vω1⊕11V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ψ1+2ψ2+2ψ3⊕V2ω1+2ψ2+2ψ3⊕Vω1+2ψ1−2ψ2+4ψ3⊕V3ω1+2ψ1⊕Vω1+2ψ1+4ψ2−2ψ3⊕V2ω1−2ψ2+4ψ3⊕V2ψ2+2ψ3⊕V4ψ1⊕V2ω1+4ψ2−2ψ3⊕Vω1−2ψ1+2ψ2+2ψ3⊕2Vω1+2ψ1⊕V−2ψ2+4ψ3⊕3V2ω1⊕V4ψ2−2ψ3⊕Vω1−2ψ1−2ψ2+4ψ3⊕Vω1+2ψ1−4ψ2+2ψ3⊕V3ω1−2ψ1⊕Vω1−2ψ1+4ψ2−2ψ3⊕Vω1+2ψ1+2ψ2−4ψ3⊕V2ω1−4ψ2+2ψ3⊕3V0⊕V2ω1+2ψ2−4ψ3⊕2Vω1−2ψ1⊕Vω1+2ψ1−2ψ2−2ψ3⊕V−4ψ2+2ψ3⊕V2ω1−2ψ2−2ψ3⊕V2ψ2−4ψ3⊕Vω1−2ψ1−4ψ2+2ψ3⊕Vω1−2ψ1+2ψ2−4ψ3⊕V−4ψ1⊕V−2ψ2−2ψ3⊕Vω1−2ψ1−2ψ2−2ψ3
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 38) ; the vectors are over the primal subalgebra.−g11+g−7g−2g6+g−1−g5+g−3−h5+h3−h6+h1h2−g3+g−5g1+g−6g2−g7+g−11g16g21g10g20g15+g4g24+g4g25g9g14g18g27+g8g19+g8g12g13g22g17g31g28g33g30g34g23g32g26g29g35g36
weight00000000000ω1ω1ω1ω1ω1ω1ω1ω1ω1ω1ω1ω1ω1ω1ω1ω12ω12ω12ω12ω12ω12ω12ω12ω12ω13ω13ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). −2ψ2−2ψ3−4ψ12ψ2−4ψ3−4ψ2+2ψ30004ψ2−2ψ3−2ψ2+4ψ34ψ12ψ2+2ψ3ω1−2ψ1−2ψ2−2ψ3ω1−2ψ1+2ψ2−4ψ3ω1−2ψ1−4ψ2+2ψ3ω1+2ψ1−2ψ2−2ψ3ω1−2ψ1ω1−2ψ1ω1+2ψ1+2ψ2−4ψ3ω1−2ψ1+4ψ2−2ψ3ω1+2ψ1−4ψ2+2ψ3ω1−2ψ1−2ψ2+4ψ3ω1+2ψ1ω1+2ψ1ω1−2ψ1+2ψ2+2ψ3ω1+2ψ1+4ψ2−2ψ3ω1+2ψ1−2ψ2+4ψ3ω1+2ψ1+2ψ2+2ψ32ω1−2ψ2−2ψ32ω1+2ψ2−4ψ32ω1−4ψ2+2ψ32ω12ω12ω12ω1+4ψ2−2ψ32ω1−2ψ2+4ψ32ω1+2ψ2+2ψ33ω1−2ψ13ω1+2ψ1
Isotypic module decomposition over primal subalgebra (total 33 isotypic components).
Isotypical components + highest weightV−2ψ2−2ψ3 → (0, 0, -2, -2)V−4ψ1 → (0, -4, 0, 0)V2ψ2−4ψ3 → (0, 0, 2, -4)V−4ψ2+2ψ3 → (0, 0, -4, 2)V0 → (0, 0, 0, 0)V4ψ2−2ψ3 → (0, 0, 4, -2)V−2ψ2+4ψ3 → (0, 0, -2, 4)V4ψ1 → (0, 4, 0, 0)V2ψ2+2ψ3 → (0, 0, 2, 2)Vω1−2ψ1−2ψ2−2ψ3 → (1, -2, -2, -2)Vω1−2ψ1+2ψ2−4ψ3 → (1, -2, 2, -4)Vω1−2ψ1−4ψ2+2ψ3 → (1, -2, -4, 2)Vω1+2ψ1−2ψ2−2ψ3 → (1, 2, -2, -2)Vω1−2ψ1 → (1, -2, 0, 0)Vω1+2ψ1+2ψ2−4ψ3 → (1, 2, 2, -4)Vω1−2ψ1+4ψ2−2ψ3 → (1, -2, 4, -2)Vω1+2ψ1−4ψ2+2ψ3 → (1, 2, -4, 2)Vω1−2ψ1−2ψ2+4ψ3 → (1, -2, -2, 4)Vω1+2ψ1 → (1, 2, 0, 0)Vω1−2ψ1+2ψ2+2ψ3 → (1, -2, 2, 2)Vω1+2ψ1+4ψ2−2ψ3 → (1, 2, 4, -2)Vω1+2ψ1−2ψ2+4ψ3 → (1, 2, -2, 4)Vω1+2ψ1+2ψ2+2ψ3 → (1, 2, 2, 2)V2ω1−2ψ2−2ψ3 → (2, 0, -2, -2)V2ω1+2ψ2−4ψ3 → (2, 0, 2, -4)V2ω1−4ψ2+2ψ3 → (2, 0, -4, 2)V2ω1 → (2, 0, 0, 0)V2ω1+4ψ2−2ψ3 → (2, 0, 4, -2)V2ω1−2ψ2+4ψ3 → (2, 0, -2, 4)V2ω1+2ψ2+2ψ3 → (2, 0, 2, 2)V3ω1−2ψ1 → (3, -2, 0, 0)V3ω1+2ψ1 → (3, 2, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13W14W15W16W17W18W19W20W21W22W23W24W25W26W27W28W29W30W31W32W33
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
−g11+g−7
g−2
g6+g−1
−g5+g−3
Cartan of centralizer component.
−h5+h3
−h6+h1
h2
−g3+g−5
g1+g−6
g2
−g7+g−11
g16
−g−17
g21
g−22
g10
−g−13
g20
g−12
g15+g4
g−8+g−19
g24+g4
g−8+g−27
g25
−g−18
g9
−g−14
g14
g−9
g18
g−25
g27+g8
−g−4−g−24
g19+g8
−g−4−g−15
g12
−g−20
g13
g−10
g22
−g−21
g17
g−16
g31
g11+g−7
−2g−29
g28
g6−g−1
−2g−26
g33
−g5−g−3
−2g−32
Semisimple subalgebra component.
−g34−g30−g23
2h6+4h5+6h4+4h3+3h2+2h1
2g−23+2g−30+2g−34
g34
−h6−2h5−2h4−2h3−h2−h1
−2g−34
g30
−h6−h5−2h4−h3−h2−h1
−2g−30
g32
−g3−g−5
−2g−33
g26
−g1+g−6
−2g−28
g29
g7+g−11
−2g−31
g35
g24+g15−g4
2g−8−2g−19−2g−27
6g−36
g36
g27+g19−g8
−2g−4+2g−15+2g−24
−6g−35
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above000000000ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		2ω1
0
−2ω1		2ω1
0
−2ω1		2ω1
0
−2ω1		2ω1
0
−2ω1		2ω1
0
−2ω1		2ω1
0
−2ω1		2ω1
0
−2ω1		2ω1
0
−2ω1		3ω1
ω1
−ω1
−3ω1		3ω1
ω1
−ω1
−3ω1
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer−2ψ2−2ψ3−4ψ12ψ2−4ψ3−4ψ2+2ψ304ψ2−2ψ3−2ψ2+4ψ34ψ12ψ2+2ψ3ω1−2ψ1−2ψ2−2ψ3
−ω1−2ψ1−2ψ2−2ψ3		ω1−2ψ1+2ψ2−4ψ3
−ω1−2ψ1+2ψ2−4ψ3		ω1−2ψ1−4ψ2+2ψ3
−ω1−2ψ1−4ψ2+2ψ3		ω1+2ψ1−2ψ2−2ψ3
−ω1+2ψ1−2ψ2−2ψ3		ω1−2ψ1
−ω1−2ψ1		ω1+2ψ1+2ψ2−4ψ3
−ω1+2ψ1+2ψ2−4ψ3		ω1−2ψ1+4ψ2−2ψ3
−ω1−2ψ1+4ψ2−2ψ3		ω1+2ψ1−4ψ2+2ψ3
−ω1+2ψ1−4ψ2+2ψ3		ω1−2ψ1−2ψ2+4ψ3
−ω1−2ψ1−2ψ2+4ψ3		ω1+2ψ1
−ω1+2ψ1		ω1−2ψ1+2ψ2+2ψ3
−ω1−2ψ1+2ψ2+2ψ3		ω1+2ψ1+4ψ2−2ψ3
−ω1+2ψ1+4ψ2−2ψ3		ω1+2ψ1−2ψ2+4ψ3
−ω1+2ψ1−2ψ2+4ψ3		ω1+2ψ1+2ψ2+2ψ3
−ω1+2ψ1+2ψ2+2ψ3		2ω1−2ψ2−2ψ3
−2ψ2−2ψ3
−2ω1−2ψ2−2ψ3		2ω1+2ψ2−4ψ3
2ψ2−4ψ3
−2ω1+2ψ2−4ψ3		2ω1−4ψ2+2ψ3
−4ψ2+2ψ3
−2ω1−4ψ2+2ψ3		2ω1
0
−2ω1		2ω1
0
−2ω1		2ω1+4ψ2−2ψ3
4ψ2−2ψ3
−2ω1+4ψ2−2ψ3		2ω1−2ψ2+4ψ3
−2ψ2+4ψ3
−2ω1−2ψ2+4ψ3		2ω1+2ψ2+2ψ3
2ψ2+2ψ3
−2ω1+2ψ2+2ψ3		3ω1−2ψ1
ω1−2ψ1
−ω1−2ψ1
−3ω1−2ψ1		3ω1+2ψ1
ω1+2ψ1
−ω1+2ψ1
−3ω1+2ψ1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M−2ψ2−2ψ3M−4ψ1M2ψ2−4ψ3M−4ψ2+2ψ3M0M4ψ2−2ψ3M−2ψ2+4ψ3M4ψ1M2ψ2+2ψ3Mω1−2ψ1−2ψ2−2ψ3⊕M−ω1−2ψ1−2ψ2−2ψ3Mω1−2ψ1+2ψ2−4ψ3⊕M−ω1−2ψ1+2ψ2−4ψ3Mω1−2ψ1−4ψ2+2ψ3⊕M−ω1−2ψ1−4ψ2+2ψ3Mω1+2ψ1−2ψ2−2ψ3⊕M−ω1+2ψ1−2ψ2−2ψ3Mω1−2ψ1⊕M−ω1−2ψ1Mω1+2ψ1+2ψ2−4ψ3⊕M−ω1+2ψ1+2ψ2−4ψ3Mω1−2ψ1+4ψ2−2ψ3⊕M−ω1−2ψ1+4ψ2−2ψ3Mω1+2ψ1−4ψ2+2ψ3⊕M−ω1+2ψ1−4ψ2+2ψ3Mω1−2ψ1−2ψ2+4ψ3⊕M−ω1−2ψ1−2ψ2+4ψ3Mω1+2ψ1⊕M−ω1+2ψ1Mω1−2ψ1+2ψ2+2ψ3⊕M−ω1−2ψ1+2ψ2+2ψ3Mω1+2ψ1+4ψ2−2ψ3⊕M−ω1+2ψ1+4ψ2−2ψ3Mω1+2ψ1−2ψ2+4ψ3⊕M−ω1+2ψ1−2ψ2+4ψ3Mω1+2ψ1+2ψ2+2ψ3⊕M−ω1+2ψ1+2ψ2+2ψ3M2ω1−2ψ2−2ψ3⊕M−2ψ2−2ψ3⊕M−2ω1−2ψ2−2ψ3M2ω1+2ψ2−4ψ3⊕M2ψ2−4ψ3⊕M−2ω1+2ψ2−4ψ3M2ω1−4ψ2+2ψ3⊕M−4ψ2+2ψ3⊕M−2ω1−4ψ2+2ψ3M2ω1⊕M0⊕M−2ω1M2ω1⊕M0⊕M−2ω1M2ω1+4ψ2−2ψ3⊕M4ψ2−2ψ3⊕M−2ω1+4ψ2−2ψ3M2ω1−2ψ2+4ψ3⊕M−2ψ2+4ψ3⊕M−2ω1−2ψ2+4ψ3M2ω1+2ψ2+2ψ3⊕M2ψ2+2ψ3⊕M−2ω1+2ψ2+2ψ3M3ω1−2ψ1⊕Mω1−2ψ1⊕M−ω1−2ψ1⊕M−3ω1−2ψ1M3ω1+2ψ1⊕Mω1+2ψ1⊕M−ω1+2ψ1⊕M−3ω1+2ψ1
Isotypic characterM−2ψ2−2ψ3M−4ψ1M2ψ2−4ψ3M−4ψ2+2ψ33M0M4ψ2−2ψ3M−2ψ2+4ψ3M4ψ1M2ψ2+2ψ3Mω1−2ψ1−2ψ2−2ψ3⊕M−ω1−2ψ1−2ψ2−2ψ3Mω1−2ψ1+2ψ2−4ψ3⊕M−ω1−2ψ1+2ψ2−4ψ3Mω1−2ψ1−4ψ2+2ψ3⊕M−ω1−2ψ1−4ψ2+2ψ3Mω1+2ψ1−2ψ2−2ψ3⊕M−ω1+2ψ1−2ψ2−2ψ32Mω1−2ψ1⊕2M−ω1−2ψ1Mω1+2ψ1+2ψ2−4ψ3⊕M−ω1+2ψ1+2ψ2−4ψ3Mω1−2ψ1+4ψ2−2ψ3⊕M−ω1−2ψ1+4ψ2−2ψ3Mω1+2ψ1−4ψ2+2ψ3⊕M−ω1+2ψ1−4ψ2+2ψ3Mω1−2ψ1−2ψ2+4ψ3⊕M−ω1−2ψ1−2ψ2+4ψ32Mω1+2ψ1⊕2M−ω1+2ψ1Mω1−2ψ1+2ψ2+2ψ3⊕M−ω1−2ψ1+2ψ2+2ψ3Mω1+2ψ1+4ψ2−2ψ3⊕M−ω1+2ψ1+4ψ2−2ψ3Mω1+2ψ1−2ψ2+4ψ3⊕M−ω1+2ψ1−2ψ2+4ψ3Mω1+2ψ1+2ψ2+2ψ3⊕M−ω1+2ψ1+2ψ2+2ψ3M2ω1−2ψ2−2ψ3⊕M−2ψ2−2ψ3⊕M−2ω1−2ψ2−2ψ3M2ω1+2ψ2−4ψ3⊕M2ψ2−4ψ3⊕M−2ω1+2ψ2−4ψ3M2ω1−4ψ2+2ψ3⊕M−4ψ2+2ψ3⊕M−2ω1−4ψ2+2ψ3M2ω1⊕M0⊕M−2ω12M2ω1⊕2M0⊕2M−2ω1M2ω1+4ψ2−2ψ3⊕M4ψ2−2ψ3⊕M−2ω1+4ψ2−2ψ3M2ω1−2ψ2+4ψ3⊕M−2ψ2+4ψ3⊕M−2ω1−2ψ2+4ψ3M2ω1+2ψ2+2ψ3⊕M2ψ2+2ψ3⊕M−2ω1+2ψ2+2ψ3M3ω1−2ψ1⊕Mω1−2ψ1⊕M−ω1−2ψ1⊕M−3ω1−2ψ1M3ω1+2ψ1⊕Mω1+2ψ1⊕M−ω1+2ψ1⊕M−3ω1+2ψ1

Semisimple subalgebra: W_{27}
Centralizer extension: W_{1}+W_{2}+W_{3}+W_{4}+W_{5}+W_{6}+W_{7}+W_{8}+W_{9}

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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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, 1.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00): (350.00, 300.00)
1: (0.00, 1.00, 0.00, 0.00): (200.00, 312.50)
2: (0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
3: (0.00, 0.00, 0.00, 1.00): (200.00, 300.00)



Made total 5853604 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_{-23}+x_{2} g_{-26}+x_{3} g_{-28}+x_{4} g_{-29}+x_{5} g_{-30}+x_{6} g_{-31}+x_{7} g_{-32}+x_{8} g_{-33} \\ +x_{9} g_{-34}, x_{18} g_{34}+x_{17} g_{33}+x_{16} g_{32}+x_{15} g_{31}+x_{14} g_{30}+x_{13} g_{29}+x_{12} g_{28}+x_{11} g_{26} \\ +x_{10} g_{23})
h: (2, 3, 4, 6, 4, 2), e = combination of g_{23} g_{26} g_{28} g_{29} g_{30} g_{31} g_{32} g_{33} g_{34} , f= combination of g_{-23} g_{-26} g_{-28} g_{-29} g_{-30} g_{-31} g_{-32} g_{-33} g_{-34} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: 2V_{3\omega_{1}}+9V_{2\omega_{1}}+18V_{\omega_{1}}+20V_{0}+18V_{-\omega_{1}}+9V_{-2\omega_{1}}+2V_{-3\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11}
+x_{1} x_{10} -3= 0
2x_{9} x_{18} +x_{8} x_{17} +2x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +2x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11}
+x_{1} x_{10} -4= 0
2x_{9} x_{18} +2x_{8} x_{17} +x_{7} x_{16} +2x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11}
+x_{1} x_{10} -4= 0
x_{8} x_{15} +x_{5} x_{12} +x_{2} x_{10} = 0
x_{7} x_{13} +x_{5} x_{11} +x_{3} x_{10} = 0
x_{9} x_{15} +x_{7} x_{12} +x_{4} x_{10} = 0
x_{9} x_{13} +x_{8} x_{11} +x_{6} x_{10} = 0
x_{6} x_{17} +x_{3} x_{14} +x_{1} x_{11} = 0
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{5} x_{14} +x_{4} x_{13} +x_{2} x_{11} -2= 0
x_{9} x_{17} +x_{7} x_{14} +x_{4} x_{11} = 0
x_{4} x_{16} +x_{2} x_{14} +x_{1} x_{12} = 0
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{3} x_{12} -2= 0
x_{9} x_{16} +x_{8} x_{14} +x_{6} x_{12} = 0
x_{6} x_{18} +x_{3} x_{16} +x_{1} x_{13} = 0
x_{8} x_{18} +x_{5} x_{16} +x_{2} x_{13} = 0
x_{4} x_{18} +x_{2} x_{17} +x_{1} x_{15} = 0
x_{7} x_{18} +x_{5} x_{17} +x_{3} x_{15} = 0
The above system after transformation.
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11}
+x_{1} x_{10} -3= 0
2x_{9} x_{18} +x_{8} x_{17} +2x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +2x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11}
+x_{1} x_{10} -4= 0
2x_{9} x_{18} +2x_{8} x_{17} +x_{7} x_{16} +2x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11}
+x_{1} x_{10} -4= 0
x_{8} x_{15} +x_{5} x_{12} +x_{2} x_{10} = 0
x_{7} x_{13} +x_{5} x_{11} +x_{3} x_{10} = 0
x_{9} x_{15} +x_{7} x_{12} +x_{4} x_{10} = 0
x_{9} x_{13} +x_{8} x_{11} +x_{6} x_{10} = 0
x_{6} x_{17} +x_{3} x_{14} +x_{1} x_{11} = 0
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{5} x_{14} +x_{4} x_{13} +x_{2} x_{11} -2= 0
x_{9} x_{17} +x_{7} x_{14} +x_{4} x_{11} = 0
x_{4} x_{16} +x_{2} x_{14} +x_{1} x_{12} = 0
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{3} x_{12} -2= 0
x_{9} x_{16} +x_{8} x_{14} +x_{6} x_{12} = 0
x_{6} x_{18} +x_{3} x_{16} +x_{1} x_{13} = 0
x_{8} x_{18} +x_{5} x_{16} +x_{2} x_{13} = 0
x_{4} x_{18} +x_{2} x_{17} +x_{1} x_{15} = 0
x_{7} x_{18} +x_{5} x_{17} +x_{3} x_{15} = 0
For the calculator:
(DynkinType =A^{3}_1; ElementsCartan =((2, 3, 4, 6, 4, 2)); generators =(x_{1} g_{-23}+x_{2} g_{-26}+x_{3} g_{-28}+x_{4} g_{-29}+x_{5} g_{-30}+x_{6} g_{-31}+x_{7} g_{-32}+x_{8} g_{-33}+x_{9} g_{-34}, x_{18} g_{34}+x_{17} g_{33}+x_{16} g_{32}+x_{15} g_{31}+x_{14} g_{30}+x_{13} g_{29}+x_{12} g_{28}+x_{11} g_{26}+x_{10} g_{23}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -3, 2x_{9} x_{18} +x_{8} x_{17} +2x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +2x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -4, 2x_{9} x_{18} +2x_{8} x_{17} +x_{7} x_{16} +2x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -4, x_{8} x_{15} +x_{5} x_{12} +x_{2} x_{10} , x_{7} x_{13} +x_{5} x_{11} +x_{3} x_{10} , x_{9} x_{15} +x_{7} x_{12} +x_{4} x_{10} , x_{9} x_{13} +x_{8} x_{11} +x_{6} x_{10} , x_{6} x_{17} +x_{3} x_{14} +x_{1} x_{11} , x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{5} x_{14} +x_{4} x_{13} +x_{2} x_{11} -2, x_{9} x_{17} +x_{7} x_{14} +x_{4} x_{11} , x_{4} x_{16} +x_{2} x_{14} +x_{1} x_{12} , x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{3} x_{12} -2, x_{9} x_{16} +x_{8} x_{14} +x_{6} x_{12} , x_{6} x_{18} +x_{3} x_{16} +x_{1} x_{13} , x_{8} x_{18} +x_{5} x_{16} +x_{2} x_{13} , x_{4} x_{18} +x_{2} x_{17} +x_{1} x_{15} , x_{7} x_{18} +x_{5} x_{17} +x_{3} x_{15} )