Subalgebra C13+2A11 ↪ C15
116 out of 119
Computations done by the calculator project.

Subalgebra type: C13+2A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from C13+A11 .
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: C13: (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2, A11: (0, 0, 0, 2, 1): 2, A11: (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 27.
Negative simple generators: g−24, g2, g19, g−13, g−5
Positive simple generators: g24, g−2, g−19, g13, g5
Cartan symmetric matrix: (1−1/2000−1/21−1000−12000002000002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4−2000−24−2000−22000002000002)
Decomposition of ambient Lie algebra: V2ω5⊕Vω4+ω5⊕Vω1+ω5⊕V2ω4⊕Vω1+ω4⊕V2ω1
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 6) ; the vectors are over the primal subalgebra.g25g20g13g17g9g5
weight2ω1ω1+ω42ω4ω1+ω5ω4+ω52ω5
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weightV2ω1 → (2, 0, 0, 0, 0)Vω1+ω4 → (1, 0, 0, 1, 0)V2ω4 → (0, 0, 0, 2, 0)Vω1+ω5 → (1, 0, 0, 0, 1)Vω4+ω5 → (0, 0, 0, 1, 1)V2ω5 → (0, 0, 0, 0, 2)
Module label W1W2W3W4W5W6
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
−g25
g1
2g−23
−g6
−2g−21
g22
4g−19
−2g−2
g24
4h5+8h4+8h3
−4h2
−2h5−4h4−4h3−4h2−2h1
4g2
−8g19
−2g−24
4g−22
−4g21
4g−6
−8g23
4g−1
8g−25
g20
−g−7
−g10
g−3
−g−18
−g16
g−16
−g18
g3
g−10
g7
g−20
Semisimple subalgebra component.
−g13
h5+2h4
2g−13
g17
−g−11
−g14
g−8
−g−15
−g12
g−12
−g15
g8
g−14
g11
g−17
g9
−g−4
−g4
−g−9
Semisimple subalgebra component.
−g5
h5
2g−5
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above2ω1
ω2
−2ω1+2ω2
ω1−ω2+ω3
−ω1+ω3
ω1+ω2−ω3
−2ω2+2ω3
−ω1+2ω2−ω3
2ω1−ω2
0
0
0
ω1−2ω2+ω3
2ω2−2ω3
−2ω1+ω2
−ω1−ω2+ω3
ω1−ω3
−ω1+ω2−ω3
2ω1−2ω2
−ω2
−2ω1		ω1+ω4
−ω1+ω2+ω4
ω1−ω4
−ω2+ω3+ω4
−ω1+ω2−ω4
ω2−ω3+ω4
−ω2+ω3−ω4
ω1−ω2+ω4
ω2−ω3−ω4
−ω1+ω4
ω1−ω2−ω4
−ω1−ω4		2ω4
0
−2ω4		ω1+ω5
−ω1+ω2+ω5
ω1−ω5
−ω2+ω3+ω5
−ω1+ω2−ω5
ω2−ω3+ω5
−ω2+ω3−ω5
ω1−ω2+ω5
ω2−ω3−ω5
−ω1+ω5
ω1−ω2−ω5
−ω1−ω5		ω4+ω5
−ω4+ω5
ω4−ω5
−ω4−ω5		2ω5
0
−2ω5
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer2ω1
ω2
−2ω1+2ω2
ω1−ω2+ω3
−ω1+ω3
ω1+ω2−ω3
−2ω2+2ω3
−ω1+2ω2−ω3
2ω1−ω2
0
0
0
ω1−2ω2+ω3
2ω2−2ω3
−2ω1+ω2
−ω1−ω2+ω3
ω1−ω3
−ω1+ω2−ω3
2ω1−2ω2
−ω2
−2ω1		ω1+ω4
−ω1+ω2+ω4
ω1−ω4
−ω2+ω3+ω4
−ω1+ω2−ω4
ω2−ω3+ω4
−ω2+ω3−ω4
ω1−ω2+ω4
ω2−ω3−ω4
−ω1+ω4
ω1−ω2−ω4
−ω1−ω4		2ω4
0
−2ω4		ω1+ω5
−ω1+ω2+ω5
ω1−ω5
−ω2+ω3+ω5
−ω1+ω2−ω5
ω2−ω3+ω5
−ω2+ω3−ω5
ω1−ω2+ω5
ω2−ω3−ω5
−ω1+ω5
ω1−ω2−ω5
−ω1−ω5		ω4+ω5
−ω4+ω5
ω4−ω5
−ω4−ω5		2ω5
0
−2ω5
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M2ω1⊕Mω1−ω2+ω3⊕Mω2⊕M2ω1−ω2⊕Mω1+ω2−ω3⊕M−2ω2+2ω3⊕M−ω1+ω3⊕Mω1−2ω2+ω3⊕M−2ω1+2ω2⊕3M0⊕M2ω1−2ω2⊕M−ω1+2ω2−ω3⊕Mω1−ω3⊕M2ω2−2ω3⊕M−ω1−ω2+ω3⊕M−2ω1+ω2⊕M−ω2⊕M−ω1+ω2−ω3⊕M−2ω1Mω1+ω4⊕M−ω2+ω3+ω4⊕M−ω1+ω2+ω4⊕Mω1−ω2+ω4⊕Mω2−ω3+ω4⊕M−ω1+ω4⊕Mω1−ω4⊕M−ω2+ω3−ω4⊕M−ω1+ω2−ω4⊕Mω1−ω2−ω4⊕Mω2−ω3−ω4⊕M−ω1−ω4M2ω4⊕M0⊕M−2ω4Mω1+ω5⊕M−ω2+ω3+ω5⊕M−ω1+ω2+ω5⊕Mω1−ω2+ω5⊕Mω2−ω3+ω5⊕M−ω1+ω5⊕Mω1−ω5⊕M−ω2+ω3−ω5⊕M−ω1+ω2−ω5⊕Mω1−ω2−ω5⊕Mω2−ω3−ω5⊕M−ω1−ω5Mω4+ω5⊕M−ω4+ω5⊕Mω4−ω5⊕M−ω4−ω5M2ω5⊕M0⊕M−2ω5
Isotypic characterM2ω1⊕Mω1−ω2+ω3⊕Mω2⊕M2ω1−ω2⊕Mω1+ω2−ω3⊕M−2ω2+2ω3⊕M−ω1+ω3⊕Mω1−2ω2+ω3⊕M−2ω1+2ω2⊕3M0⊕M2ω1−2ω2⊕M−ω1+2ω2−ω3⊕Mω1−ω3⊕M2ω2−2ω3⊕M−ω1−ω2+ω3⊕M−2ω1+ω2⊕M−ω2⊕M−ω1+ω2−ω3⊕M−2ω1Mω1+ω4⊕M−ω2+ω3+ω4⊕M−ω1+ω2+ω4⊕Mω1−ω2+ω4⊕Mω2−ω3+ω4⊕M−ω1+ω4⊕Mω1−ω4⊕M−ω2+ω3−ω4⊕M−ω1+ω2−ω4⊕Mω1−ω2−ω4⊕Mω2−ω3−ω4⊕M−ω1−ω4M2ω4⊕M0⊕M−2ω4Mω1+ω5⊕M−ω2+ω3+ω5⊕M−ω1+ω2+ω5⊕Mω1−ω2+ω5⊕Mω2−ω3+ω5⊕M−ω1+ω5⊕Mω1−ω5⊕M−ω2+ω3−ω5⊕M−ω1+ω2−ω5⊕Mω1−ω2−ω5⊕Mω2−ω3−ω5⊕M−ω1−ω5Mω4+ω5⊕M−ω4+ω5⊕Mω4−ω5⊕M−ω4−ω5M2ω5⊕M0⊕M−2ω5

Semisimple subalgebra: W_{1}+W_{3}+W_{6}
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, 367.50)
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, 1.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00): (250.00, 417.50)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (250.00, 467.50)
2: (0.00, 0.00, 1.00, 0.00, 0.00): (250.00, 467.50)
3: (0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 367.50)
4: (0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 367.50)




Made total 618 arithmetic operations while solving the Serre relations polynomial system.