Subalgebra B12+A351 ↪ C15
87 out of 119
Computations done by the calculator project.

Subalgebra type: B12+A351 (click on type for detailed printout).
Subalgebra is (parabolically) induced from B12 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: C15

Elements Cartan subalgebra scaled to act by two by components: B12: (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, A351: (0, 0, 10, 16, 9): 70
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: g−25, g1, g−3+g−4+g−5
Positive simple generators: g25, g−1, 9g5+8g4+5g3
Cartan symmetric matrix: (2−10−110002/35)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−20−2400070)
Decomposition of ambient Lie algebra: V10ω3⊕V6ω3⊕Vω2+5ω3⊕V2ω3⊕V2ω2
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.
Highest vectors of representations (total 5) ; the vectors are over the primal subalgebra.g23g5+8/9g4+5/9g3g21−g13+5/8g12g19
weight2ω22ω3ω2+5ω36ω310ω3
Isotypic module decomposition over primal subalgebra (total 5 isotypic components).
Isotypical components + highest weightV2ω2 → (0, 2, 0)V2ω3 → (0, 0, 2)Vω2+5ω3 → (0, 1, 5)V6ω3 → (0, 0, 6)V10ω3 → (0, 0, 10)
Module label W1W2W3W4W5
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
−g23
−g24
g−1
−2g25
2h1
2h5+4h4+4h3+4h2+4h1
2g−25
−2g1
−2g−24
4g−23
Semisimple subalgebra component.
−9/5g5−8/5g4−g3
9/5h5+16/5h4+2h3
2/5g−3+2/5g−4+2/5g−5
g21
g22
g18
−g−6
g20
g15
g−2
−g−10
g17
−g11
g−7
−g−14
−g14
g7
g−11
−g−17
g10
−g2
g−15
g−20
−g6
−g−18
−g−22
g−21
−g13+5/8g12
−3/8g9−5/8g8
−3/4g5−1/4g4+5/8g3
3/4h5+1/2h4−5/4h3
−3/2g−3+3/8g−4+g−5
−15/8g−8−5/8g−9
−5/4g−12+5/4g−13
g19
g16
2g13+g12
3g9−g8
6g5−4g4+g3
−6h5+8h4−2h3
−6g−3+15g−4−20g−5
−21g−8+35g−9
−56g−12−70g−13
126g−16
−252g−19
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		2ω3
0
−2ω3		ω2+5ω3
ω1−ω2+5ω3
ω2+3ω3
−ω1+ω2+5ω3
ω1−ω2+3ω3
ω2+ω3
−ω2+5ω3
−ω1+ω2+3ω3
ω1−ω2+ω3
ω2−ω3
−ω2+3ω3
−ω1+ω2+ω3
ω1−ω2−ω3
ω2−3ω3
−ω2+ω3
−ω1+ω2−ω3
ω1−ω2−3ω3
ω2−5ω3
−ω2−ω3
−ω1+ω2−3ω3
ω1−ω2−5ω3
−ω2−3ω3
−ω1+ω2−5ω3
−ω2−5ω3		6ω3
4ω3
2ω3
0
−2ω3
−4ω3
−6ω3		10ω3
8ω3
6ω3
4ω3
2ω3
0
−2ω3
−4ω3
−6ω3
−8ω3
−10ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		2ω3
0
−2ω3		ω2+5ω3
ω1−ω2+5ω3
ω2+3ω3
−ω1+ω2+5ω3
ω1−ω2+3ω3
ω2+ω3
−ω2+5ω3
−ω1+ω2+3ω3
ω1−ω2+ω3
ω2−ω3
−ω2+3ω3
−ω1+ω2+ω3
ω1−ω2−ω3
ω2−3ω3
−ω2+ω3
−ω1+ω2−ω3
ω1−ω2−3ω3
ω2−5ω3
−ω2−ω3
−ω1+ω2−3ω3
ω1−ω2−5ω3
−ω2−3ω3
−ω1+ω2−5ω3
−ω2−5ω3		6ω3
4ω3
2ω3
0
−2ω3
−4ω3
−6ω3		10ω3
8ω3
6ω3
4ω3
2ω3
0
−2ω3
−4ω3
−6ω3
−8ω3
−10ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M2ω3⊕M0⊕M−2ω3Mω2+5ω3⊕M−ω1+ω2+5ω3⊕Mω1−ω2+5ω3⊕M−ω2+5ω3⊕Mω2+3ω3⊕M−ω1+ω2+3ω3⊕Mω1−ω2+3ω3⊕M−ω2+3ω3⊕Mω2+ω3⊕M−ω1+ω2+ω3⊕Mω1−ω2+ω3⊕M−ω2+ω3⊕Mω2−ω3⊕M−ω1+ω2−ω3⊕Mω1−ω2−ω3⊕M−ω2−ω3⊕Mω2−3ω3⊕M−ω1+ω2−3ω3⊕Mω1−ω2−3ω3⊕M−ω2−3ω3⊕Mω2−5ω3⊕M−ω1+ω2−5ω3⊕Mω1−ω2−5ω3⊕M−ω2−5ω3M6ω3⊕M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3⊕M−6ω3M10ω3⊕M8ω3⊕M6ω3⊕M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3⊕M−6ω3⊕M−8ω3⊕M−10ω3
Isotypic characterM2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M2ω3⊕M0⊕M−2ω3Mω2+5ω3⊕M−ω1+ω2+5ω3⊕Mω1−ω2+5ω3⊕M−ω2+5ω3⊕Mω2+3ω3⊕M−ω1+ω2+3ω3⊕Mω1−ω2+3ω3⊕M−ω2+3ω3⊕Mω2+ω3⊕M−ω1+ω2+ω3⊕Mω1−ω2+ω3⊕M−ω2+ω3⊕Mω2−ω3⊕M−ω1+ω2−ω3⊕Mω1−ω2−ω3⊕M−ω2−ω3⊕Mω2−3ω3⊕M−ω1+ω2−3ω3⊕Mω1−ω2−3ω3⊕M−ω2−3ω3⊕Mω2−5ω3⊕M−ω1+ω2−5ω3⊕Mω1−ω2−5ω3⊕M−ω2−5ω3M6ω3⊕M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3⊕M−6ω3M10ω3⊕M8ω3⊕M6ω3⊕M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3⊕M−6ω3⊕M−8ω3⊕M−10ω3

Semisimple subalgebra: W_{1}+W_{2}
Centralizer extension: 0

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, 1.00, 0.00)
0: (1.00, 0.00, 0.00): (300.00, 470.00)
1: (0.00, 1.00, 0.00): (250.00, 470.00)
2: (0.00, 0.00, 1.00): (200.00, 420.00)




Made total 10737 arithmetic operations while solving the Serre relations polynomial system.