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Subalgebra G12 ↪ D14
15 out of 23
Computations done by the calculator project.

Subalgebra type: G12 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A31 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: D14

Elements Cartan subalgebra scaled to act by two by components: G12: (2, 3, 2, 2): 6, (-1, -1, -1, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: g−8+g−9+g−10, g11
Positive simple generators: g10+g9+g8, g−11
Cartan symmetric matrix: (2/3−1−12)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6−3−32)
Decomposition of ambient Lie algebra: Vω2⊕2Vω1
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 3) ; the vectors are over the primal subalgebra.g7+g5g6+g5g2
weightω1ω1ω2
Isotypic module decomposition over primal subalgebra (total 2 isotypic components).
Isotypical components + highest weightVω1 → (1, 0)Vω2 → (0, 1)
Module label W1W2
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g7+g5
−g−1+g−4
g10−g8
−h4+h1
2g−8−2g−10
2g4−2g1
−2g−5−2g−7
g6+g5
−g−1+g−3
g10−g9
−h3+h1
2g−9−2g−10
2g3−2g1
−2g−5−2g−6
Semisimple subalgebra component.
−g2
g12
−g7−g6+g5
2g−1+2g−3+2g−4
6g−11
−2g10−2g9−2g8
6h4+6h3+6h2+6h1
4h4+4h3+6h2+4h1
6g−8+6g−9+6g−10
−12g11
6g4+6g3+6g1
−12g−5+12g−6+12g−7
36g−12
36g−2
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as aboveω1
−ω1+ω2
2ω1−ω2
0
−2ω1+ω2
ω1−ω2
−ω1		ω2
3ω1−ω2
ω1
−ω1+ω2
−3ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
3ω1−2ω2
ω1−ω2
−ω1
−3ω1+ω2
−ω2
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizerω1
−ω1+ω2
2ω1−ω2
0
−2ω1+ω2
ω1−ω2
−ω1		ω2
3ω1−ω2
ω1
−ω1+ω2
−3ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
3ω1−2ω2
ω1−ω2
−ω1
−3ω1+ω2
−ω2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.Mω1⊕M2ω1−ω2⊕M−ω1+ω2⊕M0⊕Mω1−ω2⊕M−2ω1+ω2⊕M−ω1M3ω1−ω2⊕Mω2⊕Mω1⊕M2ω1−ω2⊕M3ω1−2ω2⊕M−ω1+ω2⊕2M0⊕Mω1−ω2⊕M−3ω1+2ω2⊕M−2ω1+ω2⊕M−ω1⊕M−ω2⊕M−3ω1+ω2
Isotypic character2Mω1⊕2M2ω1−ω2⊕2M−ω1+ω2⊕2M0⊕2Mω1−ω2⊕2M−2ω1+ω2⊕2M−ω1M3ω1−ω2⊕Mω2⊕Mω1⊕M2ω1−ω2⊕M3ω1−2ω2⊕M−ω1+ω2⊕2M0⊕Mω1−ω2⊕M−3ω1+2ω2⊕M−2ω1+ω2⊕M−ω1⊕M−ω2⊕M−3ω1+ω2

Semisimple subalgebra: 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, 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): (266.67, 400.00)
1: (0.00, 1.00): (300.00, 500.00)




Made total 545 arithmetic operations while solving the Serre relations polynomial system.