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Subalgebra B12+A21 ↪ B14
38 out of 48
Computations done by the calculator project.

Subalgebra type: B12+A21 (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: B14

Elements Cartan subalgebra scaled to act by two by components: B12: (1, 2, 2, 2): 2, (0, -2, -2, -2): 4, A21: (0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: g−16, g12+g6, −g−3+g−10
Positive simple generators: g16, g−6+g−12, g10−g3
Cartan symmetric matrix: (2−10−110001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−20−240004)
Decomposition of ambient Lie algebra: Vω1+2ω3⊕2V2ω3⊕V2ω2⊕Vω1
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.g11g1−g10+g3g7g15
weightω12ω22ω32ω3ω1+2ω3
Isotypic module decomposition over primal subalgebra (total 5 isotypic components).
Isotypical components + highest weightVω1 → (1, 0, 0)V2ω2 → (0, 2, 0)V2ω3 → (0, 0, 2)Vω1+2ω3 → (1, 0, 2)
Module label W1W2W3W4W5
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g11
−g−9
−g4+g−4
−2g9
−2g−11
Semisimple subalgebra component.
−g1
g13+g8
−g−6−g−12
2g16
−2h4−2h3−2h2
−4h4−4h3−4h2−2h1
−2g−16
2g12+2g6
2g−8+2g−13
4g−1
Semisimple subalgebra component.
g10−g3
−2h4−2h3
2g−3−2g−10
g7
−g4−g−4
−2g−7
g15
−g−2
−g13+g8
g10+g3
g−6−g−12
2g5
−2g14
−2h4
−2g−14
−2g−5
2g12−2g6
−2g−3−2g−10
2g−8−2g−13
−4g2
−4g−15
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as aboveω1
−ω1+2ω2
0
ω1−2ω2
−ω1		2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		2ω3
0
−2ω3		2ω3
0
−2ω3		ω1+2ω3
−ω1+2ω2+2ω3
ω1
2ω3
−ω1+2ω2
ω1−2ω3
ω1−2ω2+2ω3
0
−ω1+2ω2−2ω3
−ω1+2ω3
ω1−2ω2
−2ω3
−ω1
ω1−2ω2−2ω3
−ω1−2ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizerω1
−ω1+2ω2
0
ω1−2ω2
−ω1		2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		2ω3
0
−2ω3		2ω3
0
−2ω3		ω1+2ω3
−ω1+2ω2+2ω3
ω1
2ω3
−ω1+2ω2
ω1−2ω3
ω1−2ω2+2ω3
0
−ω1+2ω2−2ω3
−ω1+2ω3
ω1−2ω2
−2ω3
−ω1
ω1−2ω2−2ω3
−ω1−2ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2M2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M2ω3⊕M0⊕M−2ω3M2ω3⊕M0⊕M−2ω3M−ω1+2ω2+2ω3⊕Mω1+2ω3⊕M2ω3⊕M−ω1+2ω3⊕Mω1−2ω2+2ω3⊕M−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2⊕M−ω1+2ω2−2ω3⊕Mω1−2ω3⊕M−2ω3⊕M−ω1−2ω3⊕Mω1−2ω2−2ω3
Isotypic characterM−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2M2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M2ω3⊕M0⊕M−2ω3M2ω3⊕M0⊕M−2ω3M−ω1+2ω2+2ω3⊕Mω1+2ω3⊕M2ω3⊕M−ω1+2ω3⊕Mω1−2ω2+2ω3⊕M−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2⊕M−ω1+2ω2−2ω3⊕Mω1−2ω3⊕M−2ω3⊕M−ω1−2ω3⊕Mω1−2ω2−2ω3

Semisimple subalgebra: W_{2}+W_{3}
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, 300.00)
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, 350.00)
1: (0.00, 1.00, 0.00): (250.00, 350.00)
2: (0.00, 0.00, 1.00): (200.00, 300.00)




Made total 57977 arithmetic operations while solving the Serre relations polynomial system.