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Subalgebra B12+A12A16
54 out of 61
Computations done by the calculator project.

Subalgebra type: B12+A12 (click on type for detailed printout).
Subalgebra is (parabolically) induced from B12+A11 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A16
Basis of Cartan of centralizer: 1 vectors: (3, 6, 2, -2, -6, -3)

Elements Cartan subalgebra scaled to act by two by components: B12: (1, 1, 1, 1, 1, 1): 2, (-1, 0, 0, 0, 0, -1): 4, A12: (0, 0, 1, 1, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 18.
Negative simple generators: g21, g6+g1, g9, g4
Positive simple generators: g21, g1+g6, g9, g4
Cartan symmetric matrix: (2100110000210012)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2200240000210012)
Decomposition of ambient Lie algebra: Vω3+ω4Vω2+ω4Vω2+ω3V2ω2Vω1V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω2+ω4+14ψVω3+ω4V2ω2Vω1V0Vω2+ω314ψ
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 6) ; the vectors are over the primal subalgebra.h62h52/3h4+2/3h3+2h2+h1g20+g19g17g14g8g3
weight0ω12ω2ω2+ω3ω2+ω4ω3+ω4
weights rel. to Cartan of (centralizer+semisimple s.a.). 0ω12ω2ω2+ω314ψω2+ω4+14ψω3+ω4
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0, 0)Vω1 → (1, 0, 0, 0, 0)V2ω2 → (0, 2, 0, 0, 0)Vω2+ω314ψ → (0, 1, 1, 0, -14)Vω2+ω4+14ψ → (0, 1, 0, 1, 14)Vω3+ω4 → (0, 0, 1, 1, 0)
Module label W1W2W3W4W5W6
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
h62h52/3h4+2/3h3+2h2+h1
g20+g19
g1+g6
h6h1
2g6+2g1
2g19+2g20
Semisimple subalgebra component.
g17
g20g19
g1g6
2g21
h6h1
2h62h52h42h32h22h1
2g21
2g6+2g1
2g19+2g20
4g17
g14
g18
g5
g7
g11
g10
g2
g16
g15
g13
g12
g8
g8
g12
g13
g15
g16
g2
g10
g11
g7
g5
g18
g14
Semisimple subalgebra component.
g3
g4
g9
h4
h4h3
g9
2g4
g3
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1
ω1+2ω2
0
ω12ω2
ω1
2ω2
ω1
ω1+2ω2
2ω12ω2
0
0
2ω1+2ω2
ω12ω2
ω1
2ω2
ω2+ω3
ω1ω2+ω3
ω2ω3+ω4
ω1+ω2+ω3
ω1ω2ω3+ω4
ω2ω4
ω2+ω3
ω1+ω2ω3+ω4
ω1ω2ω4
ω2ω3+ω4
ω1+ω2ω4
ω2ω4
ω2+ω4
ω1ω2+ω4
ω2+ω3ω4
ω1+ω2+ω4
ω1ω2+ω3ω4
ω2ω3
ω2+ω4
ω1+ω2+ω3ω4
ω1ω2ω3
ω2+ω3ω4
ω1+ω2ω3
ω2ω3
ω3+ω4
ω3+2ω4
2ω3ω4
0
0
2ω3+ω4
ω32ω4
ω3ω4
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1
ω1+2ω2
0
ω12ω2
ω1
2ω2
ω1
ω1+2ω2
2ω12ω2
0
0
2ω1+2ω2
ω12ω2
ω1
2ω2
ω2+ω314ψ
ω1ω2+ω314ψ
ω2ω3+ω414ψ
ω1+ω2+ω314ψ
ω1ω2ω3+ω414ψ
ω2ω414ψ
ω2+ω314ψ
ω1+ω2ω3+ω414ψ
ω1ω2ω414ψ
ω2ω3+ω414ψ
ω1+ω2ω414ψ
ω2ω414ψ
ω2+ω4+14ψ
ω1ω2+ω4+14ψ
ω2+ω3ω4+14ψ
ω1+ω2+ω4+14ψ
ω1ω2+ω3ω4+14ψ
ω2ω3+14ψ
ω2+ω4+14ψ
ω1+ω2+ω3ω4+14ψ
ω1ω2ω3+14ψ
ω2+ω3ω4+14ψ
ω1+ω2ω3+14ψ
ω2ω3+14ψ
ω3+ω4
ω3+2ω4
2ω3ω4
0
0
2ω3+ω4
ω32ω4
ω3ω4
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0Mω1+2ω2Mω1M0Mω1Mω12ω2M2ω2Mω1+2ω2Mω1M2ω1+2ω22M0M2ω12ω2Mω1Mω12ω2M2ω2Mω2+ω314ψMω2ω3+ω414ψMω1+ω2+ω314ψMω1ω2+ω314ψMω1+ω2ω3+ω414ψMω1ω2ω3+ω414ψMω2+ω314ψMω2ω414ψMω2ω3+ω414ψMω1+ω2ω414ψMω1ω2ω414ψMω2ω414ψMω2+ω4+14ψMω1+ω2+ω4+14ψMω1ω2+ω4+14ψMω2+ω3ω4+14ψMω2+ω4+14ψMω2ω3+14ψMω1+ω2+ω3ω4+14ψMω1ω2+ω3ω4+14ψMω1+ω2ω3+14ψMω1ω2ω3+14ψMω2+ω3ω4+14ψMω2ω3+14ψMω3+ω4Mω3+2ω4M2ω3ω42M0M2ω3+ω4Mω32ω4Mω3ω4
Isotypic characterM0Mω1+2ω2Mω1M0Mω1Mω12ω2M2ω2Mω1+2ω2Mω1M2ω1+2ω22M0M2ω12ω2Mω1Mω12ω2M2ω2Mω2+ω314ψMω2ω3+ω414ψMω1+ω2+ω314ψMω1ω2+ω314ψMω1+ω2ω3+ω414ψMω1ω2ω3+ω414ψMω2+ω314ψMω2ω414ψMω2ω3+ω414ψMω1+ω2ω414ψMω1ω2ω414ψMω2ω414ψMω2+ω4+14ψMω1+ω2+ω4+14ψMω1ω2+ω4+14ψMω2+ω3ω4+14ψMω2+ω4+14ψMω2ω3+14ψMω1+ω2+ω3ω4+14ψMω1ω2+ω3ω4+14ψMω1+ω2ω3+14ψMω1ω2ω3+14ψMω2+ω3ω4+14ψMω2ω3+14ψMω3+ω4Mω3+2ω4M2ω3ω42M0M2ω3+ω4Mω32ω4Mω3ω4

Semisimple subalgebra: W_{3}+W_{6}
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.
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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)
(0.00, 1.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00): (300.00, 350.00)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (250.00, 350.00)
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 626 arithmetic operations while solving the Serre relations polynomial system.