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Subalgebra A13+2A11E16
113 out of 119
Computations done by the calculator project.

Subalgebra type: A13+2A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A13+A11 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 1 vectors: (1, 0, 2, 0, -2, -1)

Elements Cartan subalgebra scaled to act by two by components: A13: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, A11: (0, 0, 0, 0, 0, 1): 2, A11: (1, 0, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 21.
Negative simple generators: g36, g2, g4, g6, g1
Positive simple generators: g36, g2, g4, g6, g1
Cartan symmetric matrix: (2100012100012000002000002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2100012100012000002000002)
Decomposition of ambient Lie algebra: Vω2+ω4+ω5V2ω5Vω3+ω5Vω1+ω5V2ω4Vω3+ω4Vω1+ω4Vω1+ω3V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω3+ω5+6ψVω1+ω4+6ψVω2+ω4+ω5V2ω5V2ω4Vω1+ω3V0Vω1+ω56ψVω3+ω46ψ
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 9) ; the vectors are over the primal subalgebra.h62h5+2h3+h1g34g32g11g6g33g7g1g24
weight0ω1+ω3ω1+ω4ω3+ω42ω4ω1+ω5ω3+ω52ω5ω2+ω4+ω5
weights rel. to Cartan of (centralizer+semisimple s.a.). 0ω1+ω3ω1+ω4+6ψω3+ω46ψ2ω4ω1+ω56ψω3+ω5+6ψ2ω5ω2+ω4+ω5
Isotypic module decomposition over primal subalgebra (total 9 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0, 0, 0)Vω1+ω3 → (1, 0, 1, 0, 0, 0)Vω1+ω4+6ψ → (1, 0, 0, 1, 0, 6)Vω3+ω46ψ → (0, 0, 1, 1, 0, -6)V2ω4 → (0, 0, 0, 2, 0, 0)Vω1+ω56ψ → (1, 0, 0, 0, 1, -6)Vω3+ω5+6ψ → (0, 0, 1, 0, 1, 6)V2ω5 → (0, 0, 0, 0, 2, 0)Vω2+ω4+ω5 → (0, 1, 0, 1, 1, 0)
Module label W1W2W3W4W5W6W7W8W9
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
h62h5+2h3+h1
Semisimple subalgebra component.
g34
g8
g35
g4
g2
g36
h4
h2
h6+2h5+3h4+2h3+2h2+h1
g2
2g4
g36
g35
g8
g34
g32
g14
g29
g10
g20
g5
g16
g11
g11
g16
g5
g20
g10
g29
g14
g32
Semisimple subalgebra component.
g6
h6
2g6
g33
g13
g31
g9
g17
g3
g12
g7
g7
g12
g3
g17
g9
g31
g13
g33
Semisimple subalgebra component.
g1
h1
2g1
g24
g27
g18
g21
g23
g30
g22
g25
g15
g19
g28
g26
g26
g28
g19
g15
g25
g22
g30
g23
g21
g18
g27
g24
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1+ω3
ω1+ω2+ω3
ω1+ω2ω3
ω2+2ω3
ω1+2ω2ω3
2ω1ω2
0
0
0
ω12ω2+ω3
ω22ω3
2ω1+ω2
ω1ω2+ω3
ω1ω2ω3
ω1ω3
ω1+ω4
ω1+ω2+ω4
ω1ω4
ω2+ω3+ω4
ω1+ω2ω4
ω3+ω4
ω2+ω3ω4
ω3ω4
ω3+ω4
ω2ω3+ω4
ω3ω4
ω1ω2+ω4
ω2ω3ω4
ω1+ω4
ω1ω2ω4
ω1ω4
2ω4
0
2ω4
ω1+ω5
ω1+ω2+ω5
ω1ω5
ω2+ω3+ω5
ω1+ω2ω5
ω3+ω5
ω2+ω3ω5
ω3ω5
ω3+ω5
ω2ω3+ω5
ω3ω5
ω1ω2+ω5
ω2ω3ω5
ω1+ω5
ω1ω2ω5
ω1ω5
2ω5
0
2ω5
ω2+ω4+ω5
ω1ω2+ω3+ω4+ω5
ω2ω4+ω5
ω2+ω4ω5
ω1+ω3+ω4+ω5
ω1ω3+ω4+ω5
ω1ω2+ω3ω4+ω5
ω1ω2+ω3+ω4ω5
ω2ω4ω5
ω1+ω2ω3+ω4+ω5
ω1+ω3ω4+ω5
ω1+ω3+ω4ω5
ω1ω3ω4+ω5
ω1ω3+ω4ω5
ω1ω2+ω3ω4ω5
ω2+ω4+ω5
ω1+ω2ω3ω4+ω5
ω1+ω2ω3+ω4ω5
ω1+ω3ω4ω5
ω1ω3ω4ω5
ω2ω4+ω5
ω2+ω4ω5
ω1+ω2ω3ω4ω5
ω2ω4ω5
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1+ω3
ω1+ω2+ω3
ω1+ω2ω3
ω2+2ω3
ω1+2ω2ω3
2ω1ω2
0
0
0
ω12ω2+ω3
ω22ω3
2ω1+ω2
ω1ω2+ω3
ω1ω2ω3
ω1ω3
ω1+ω4+6ψ
ω1+ω2+ω4+6ψ
ω1ω4+6ψ
ω2+ω3+ω4+6ψ
ω1+ω2ω4+6ψ
ω3+ω4+6ψ
ω2+ω3ω4+6ψ
ω3ω4+6ψ
ω3+ω46ψ
ω2ω3+ω46ψ
ω3ω46ψ
ω1ω2+ω46ψ
ω2ω3ω46ψ
ω1+ω46ψ
ω1ω2ω46ψ
ω1ω46ψ
2ω4
0
2ω4
ω1+ω56ψ
ω1+ω2+ω56ψ
ω1ω56ψ
ω2+ω3+ω56ψ
ω1+ω2ω56ψ
ω3+ω56ψ
ω2+ω3ω56ψ
ω3ω56ψ
ω3+ω5+6ψ
ω2ω3+ω5+6ψ
ω3ω5+6ψ
ω1ω2+ω5+6ψ
ω2ω3ω5+6ψ
ω1+ω5+6ψ
ω1ω2ω5+6ψ
ω1ω5+6ψ
2ω5
0
2ω5
ω2+ω4+ω5
ω1ω2+ω3+ω4+ω5
ω2ω4+ω5
ω2+ω4ω5
ω1+ω3+ω4+ω5
ω1ω3+ω4+ω5
ω1ω2+ω3ω4+ω5
ω1ω2+ω3+ω4ω5
ω2ω4ω5
ω1+ω2ω3+ω4+ω5
ω1+ω3ω4+ω5
ω1+ω3+ω4ω5
ω1ω3ω4+ω5
ω1ω3+ω4ω5
ω1ω2+ω3ω4ω5
ω2+ω4+ω5
ω1+ω2ω3ω4+ω5
ω1+ω2ω3+ω4ω5
ω1+ω3ω4ω5
ω1ω3ω4ω5
ω2ω4+ω5
ω2+ω4ω5
ω1+ω2ω3ω4ω5
ω2ω4ω5
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0Mω1+ω3Mω2+2ω3Mω1+ω2+ω3M2ω1ω2Mω1+ω2ω3Mω12ω2+ω33M0Mω1+2ω2ω3Mω1ω2+ω3M2ω1+ω2Mω1ω2ω3Mω22ω3Mω1ω3Mω1+ω4+6ψMω2+ω3+ω4+6ψMω1+ω2+ω4+6ψMω3+ω4+6ψMω1ω4+6ψMω2+ω3ω4+6ψMω1+ω2ω4+6ψMω3ω4+6ψMω3+ω46ψMω1ω2+ω46ψMω2ω3+ω46ψMω1+ω46ψMω3ω46ψMω1ω2ω46ψMω2ω3ω46ψMω1ω46ψM2ω4M0M2ω4Mω1+ω56ψMω2+ω3+ω56ψMω1+ω2+ω56ψMω3+ω56ψMω1ω56ψMω2+ω3ω56ψMω1+ω2ω56ψMω3ω56ψMω3+ω5+6ψMω1ω2+ω5+6ψMω2ω3+ω5+6ψMω1+ω5+6ψMω3ω5+6ψMω1ω2ω5+6ψMω2ω3ω5+6ψMω1ω5+6ψM2ω5M0M2ω5Mω1ω2+ω3+ω4+ω5Mω2+ω4+ω5Mω1+ω3+ω4+ω5Mω1ω3+ω4+ω5Mω2+ω4+ω5Mω1+ω2ω3+ω4+ω5Mω1ω2+ω3ω4+ω5Mω2ω4+ω5Mω1ω2+ω3+ω4ω5Mω2+ω4ω5Mω1+ω3ω4+ω5Mω1ω3ω4+ω5Mω1+ω3+ω4ω5Mω1ω3+ω4ω5Mω2ω4+ω5Mω1+ω2ω3ω4+ω5Mω2+ω4ω5Mω1+ω2ω3+ω4ω5Mω1ω2+ω3ω4ω5Mω2ω4ω5Mω1+ω3ω4ω5Mω1ω3ω4ω5Mω2ω4ω5Mω1+ω2ω3ω4ω5
Isotypic characterM0Mω1+ω3Mω2+2ω3Mω1+ω2+ω3M2ω1ω2Mω1+ω2ω3Mω12ω2+ω33M0Mω1+2ω2ω3Mω1ω2+ω3M2ω1+ω2Mω1ω2ω3Mω22ω3Mω1ω3Mω1+ω4+6ψMω2+ω3+ω4+6ψMω1+ω2+ω4+6ψMω3+ω4+6ψMω1ω4+6ψMω2+ω3ω4+6ψMω1+ω2ω4+6ψMω3ω4+6ψMω3+ω46ψMω1ω2+ω46ψMω2ω3+ω46ψMω1+ω46ψMω3ω46ψMω1ω2ω46ψMω2ω3ω46ψMω1ω46ψM2ω4M0M2ω4Mω1+ω56ψMω2+ω3+ω56ψMω1+ω2+ω56ψMω3+ω56ψMω1ω56ψMω2+ω3ω56ψMω1+ω2ω56ψMω3ω56ψMω3+ω5+6ψMω1ω2+ω5+6ψMω2ω3+ω5+6ψMω1+ω5+6ψMω3ω5+6ψMω1ω2ω5+6ψMω2ω3ω5+6ψMω1ω5+6ψM2ω5M0M2ω5Mω1ω2+ω3+ω4+ω5Mω2+ω4+ω5Mω1+ω3+ω4+ω5Mω1ω3+ω4+ω5Mω2+ω4+ω5Mω1+ω2ω3+ω4+ω5Mω1ω2+ω3ω4+ω5Mω2ω4+ω5Mω1ω2+ω3+ω4ω5Mω2+ω4ω5Mω1+ω3ω4+ω5Mω1ω3ω4+ω5Mω1+ω3+ω4ω5Mω1ω3+ω4ω5Mω2ω4+ω5Mω1+ω2ω3ω4+ω5Mω2+ω4ω5Mω1+ω2ω3+ω4ω5Mω1ω2+ω3ω4ω5Mω2ω4ω5Mω1+ω3ω4ω5Mω1ω3ω4ω5Mω2ω4ω5Mω1+ω2ω3ω4ω5

Semisimple subalgebra: W_{2}+W_{5}+W_{8}
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, 420.00)
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)
(0.00, 1.00, 0.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00, 0.00): (275.00, 470.00)
1: (0.00, 1.00, 0.00, 0.00, 0.00, 0.00): (250.00, 520.00)
2: (0.00, 0.00, 1.00, 0.00, 0.00, 0.00): (225.00, 470.00)
3: (0.00, 0.00, 0.00, 1.00, 0.00, 0.00): (200.00, 420.00)
4: (0.00, 0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 420.00)
5: (0.00, 0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 420.00)



Made total 614 arithmetic operations while solving the Serre relations polynomial system.