Subalgebra A351F14
12 out of 59
Computations done by the calculator project.

Subalgebra type: A351 (click on type for detailed printout).
Centralizer: A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: C13
Basis of Cartan of centralizer: 1 vectors: (0, 1, 0, 0)
Contained up to conjugation as a direct summand of: A351+A11 .

Elements Cartan subalgebra scaled to act by two by components: A351: (10, 19, 28, 16): 70
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g4+g8+g9
Positive simple generators: 9g9+5g8+8g4
Cartan symmetric matrix: (2/35)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (70)
Decomposition of ambient Lie algebra: V10ω12V9ω1V6ω12V3ω1V2ω13V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V9ω1+2ψV10ω1V9ω12ψV6ω1V3ω1+2ψV4ψV2ω1V3ω12ψV0V4ψ
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 10) ; the vectors are over the primal subalgebra.g2h2g2g9+5/9g8+8/9g4g11+8/5g7g14+8/5g10g19+8/5g16g22g23g24
weight0002ω13ω13ω16ω19ω19ω110ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). 4ψ04ψ2ω13ω12ψ3ω1+2ψ6ω19ω12ψ9ω1+2ψ10ω1
Isotypic module decomposition over primal subalgebra (total 10 isotypic components).
Isotypical components + highest weightV4ψ → (0, -4)V0 → (0, 0)V4ψ → (0, 4)V2ω1 → (2, 0)V3ω12ψ → (3, -2)V3ω1+2ψ → (3, 2)V6ω1 → (6, 0)V9ω12ψ → (9, -2)V9ω1+2ψ → (9, 2)V10ω1 → (10, 0)
Module label W1W2W3W4W5W6W7W8W9W10
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g2
Cartan of centralizer component.
h2
g2
Semisimple subalgebra component.
9/8g95/8g8g4
2h4+7/2h3+19/8h2+5/4h1
1/4g4+1/4g8+1/4g9
g11+8/5g7
3/5g3+g1
6/5g5+2/5g6
2/5g102/5g14
g14+8/5g10
3/5g6g5
6/5g12/5g3
2/5g7+2/5g11
g19+8/5g16
3/5g13g12
6/5g9+g82/5g4
4/5h4+2/5h34/5h22h1
3/5g412/5g8+8/5g9
3g12+g13
2g16+2g19
g22
g18
g15
2g11+g7
3g32g1
6g55g6
5g1016g14
21g17
42g20
42g23
g23
g20
g17
2g14+g10
3g6+2g5
6g1+5g3
5g7+16g11
21g15
42g18
42g22
g24
g21
g19+2g16
3g13+g12
6g9g8+4g4
8h4+14h3+8h2+2h1
15g4+6g8+20g9
21g12+35g13
70g1656g19
126g21
252g24
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0002ω1
0
2ω1
3ω1
ω1
ω1
3ω1
3ω1
ω1
ω1
3ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
9ω1
7ω1
5ω1
3ω1
ω1
ω1
3ω1
5ω1
7ω1
9ω1
9ω1
7ω1
5ω1
3ω1
ω1
ω1
3ω1
5ω1
7ω1
9ω1
10ω1
8ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
8ω1
10ω1
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer4ψ04ψ2ω1
0
2ω1
3ω12ψ
ω12ψ
ω12ψ
3ω12ψ
3ω1+2ψ
ω1+2ψ
ω1+2ψ
3ω1+2ψ
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
9ω12ψ
7ω12ψ
5ω12ψ
3ω12ψ
ω12ψ
ω12ψ
3ω12ψ
5ω12ψ
7ω12ψ
9ω12ψ
9ω1+2ψ
7ω1+2ψ
5ω1+2ψ
3ω1+2ψ
ω1+2ψ
ω1+2ψ
3ω1+2ψ
5ω1+2ψ
7ω1+2ψ
9ω1+2ψ
10ω1
8ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
8ω1
10ω1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M4ψM0M4ψM2ω1M0M2ω1M3ω12ψMω12ψMω12ψM3ω12ψM3ω1+2ψMω1+2ψMω1+2ψM3ω1+2ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M9ω12ψM7ω12ψM5ω12ψM3ω12ψMω12ψMω12ψM3ω12ψM5ω12ψM7ω12ψM9ω12ψM9ω1+2ψM7ω1+2ψM5ω1+2ψM3ω1+2ψMω1+2ψMω1+2ψM3ω1+2ψM5ω1+2ψM7ω1+2ψM9ω1+2ψM10ω1M8ω1M6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M8ω1M10ω1
Isotypic characterM4ψM0M4ψM2ω1M0M2ω1M3ω12ψMω12ψMω12ψM3ω12ψM3ω1+2ψMω1+2ψMω1+2ψM3ω1+2ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M9ω12ψM7ω12ψM5ω12ψM3ω12ψMω12ψMω12ψM3ω12ψM5ω12ψM7ω12ψM9ω12ψM9ω1+2ψM7ω1+2ψM5ω1+2ψM3ω1+2ψMω1+2ψMω1+2ψM3ω1+2ψM5ω1+2ψM7ω1+2ψM9ω1+2ψM10ω1M8ω1M6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M8ω1M10ω1

Semisimple subalgebra: W_{4}
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.
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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, 1.00)
0: (1.00, 0.00): (1950.00, 300.00)
1: (0.00, 1.00): (200.00, 312.50)



Made total 7681 arithmetic operations while solving the Serre relations polynomial system.