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Subalgebra A111+A101 ↪ C15
48 out of 119
Computations done by the calculator project.

Subalgebra type: A111+A101 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A111 .
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: A111: (6, 8, 10, 10, 5): 22, A101: (0, 0, 0, 6, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−1+g−19+g−23, g−4+g−5
Positive simple generators: 4g23+g19+3g1, 4g5+3g4
Cartan symmetric matrix: (2/11001/5)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (220020)
Decomposition of ambient Lie algebra: V6ω2⊕V3ω1+3ω2⊕V6ω1⊕Vω1+3ω2⊕V4ω1⊕V2ω2⊕3V2ω1
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 9) ; the vectors are over the primal subalgebra.g21+3g6g23+3/4g1g19g5+3/4g4g22g16g25g20g13
weight2ω12ω12ω12ω24ω1ω1+3ω26ω13ω1+3ω26ω2
Isotypic module decomposition over primal subalgebra (total 8 isotypic components).
Isotypical components + highest weightV2ω1 → (2, 0)V2ω2 → (0, 2)V4ω1 → (4, 0)Vω1+3ω2 → (1, 3)V6ω1 → (6, 0)V3ω1+3ω2 → (3, 3)V6ω2 → (0, 6)
Module label W1W2W3W4W5W6W7W8
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
4g23+g19+3g1
−5h5−10h4−10h3−8h2−6h1
−2g−1−2g−19−2g−23
g23+3/4g1
−h5−2h4−2h3−2h2−3/2h1
−1/2g−1−1/2g−23
g21+3g6
2g2−g−2
g−6+g−21
Semisimple subalgebra component.
−4/3g5−g4
4/3h5+2h4
2/3g−4+2/3g−5
g22
g21−g6
−2g2−g−2
g−6−3g−21
4g−22
g16
−g−3
g12
−g−8
−g8
−g−12
g3
g−16
g25
g24
2g23−g1
−2h5−4h4−4h3−4h2+2h1
4g−1−6g−23
10g−24
−20g−25
g20
g18
g17
−g−7
g15
−g14
g−10
−g−11
−g11
g10
g−14
−g−15
g7
g−17
g−18
−g−20
g13
g9
2g5−g4
−2h5+2h4
4g−4−6g−5
10g−9
−20g−13
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above2ω1
0
−2ω1		2ω1
0
−2ω1		2ω2
0
−2ω2		4ω1
2ω1
0
−2ω1
−4ω1		ω1+3ω2
−ω1+3ω2
ω1+ω2
−ω1+ω2
ω1−ω2
−ω1−ω2
ω1−3ω2
−ω1−3ω2		6ω1
4ω1
2ω1
0
−2ω1
−4ω1
−6ω1		3ω1+3ω2
ω1+3ω2
3ω1+ω2
−ω1+3ω2
ω1+ω2
3ω1−ω2
−3ω1+3ω2
−ω1+ω2
ω1−ω2
3ω1−3ω2
−3ω1+ω2
−ω1−ω2
ω1−3ω2
−3ω1−ω2
−ω1−3ω2
−3ω1−3ω2		6ω2
4ω2
2ω2
0
−2ω2
−4ω2
−6ω2
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer2ω1
0
−2ω1		2ω1
0
−2ω1		2ω2
0
−2ω2		4ω1
2ω1
0
−2ω1
−4ω1		ω1+3ω2
−ω1+3ω2
ω1+ω2
−ω1+ω2
ω1−ω2
−ω1−ω2
ω1−3ω2
−ω1−3ω2		6ω1
4ω1
2ω1
0
−2ω1
−4ω1
−6ω1		3ω1+3ω2
ω1+3ω2
3ω1+ω2
−ω1+3ω2
ω1+ω2
3ω1−ω2
−3ω1+3ω2
−ω1+ω2
ω1−ω2
3ω1−3ω2
−3ω1+ω2
−ω1−ω2
ω1−3ω2
−3ω1−ω2
−ω1−3ω2
−3ω1−3ω2		6ω2
4ω2
2ω2
0
−2ω2
−4ω2
−6ω2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M2ω1⊕M0⊕M−2ω1M2ω1⊕M0⊕M−2ω1M2ω2⊕M0⊕M−2ω2M4ω1⊕M2ω1⊕M0⊕M−2ω1⊕M−4ω1Mω1+3ω2⊕M−ω1+3ω2⊕Mω1+ω2⊕M−ω1+ω2⊕Mω1−ω2⊕M−ω1−ω2⊕Mω1−3ω2⊕M−ω1−3ω2M6ω1⊕M4ω1⊕M2ω1⊕M0⊕M−2ω1⊕M−4ω1⊕M−6ω1M3ω1+3ω2⊕Mω1+3ω2⊕M3ω1+ω2⊕M−ω1+3ω2⊕Mω1+ω2⊕M3ω1−ω2⊕M−3ω1+3ω2⊕M−ω1+ω2⊕Mω1−ω2⊕M3ω1−3ω2⊕M−3ω1+ω2⊕M−ω1−ω2⊕Mω1−3ω2⊕M−3ω1−ω2⊕M−ω1−3ω2⊕M−3ω1−3ω2M6ω2⊕M4ω2⊕M2ω2⊕M0⊕M−2ω2⊕M−4ω2⊕M−6ω2
Isotypic characterM2ω1⊕M0⊕M−2ω12M2ω1⊕2M0⊕2M−2ω1M2ω2⊕M0⊕M−2ω2M4ω1⊕M2ω1⊕M0⊕M−2ω1⊕M−4ω1Mω1+3ω2⊕M−ω1+3ω2⊕Mω1+ω2⊕M−ω1+ω2⊕Mω1−ω2⊕M−ω1−ω2⊕Mω1−3ω2⊕M−ω1−3ω2M6ω1⊕M4ω1⊕M2ω1⊕M0⊕M−2ω1⊕M−4ω1⊕M−6ω1M3ω1+3ω2⊕Mω1+3ω2⊕M3ω1+ω2⊕M−ω1+3ω2⊕Mω1+ω2⊕M3ω1−ω2⊕M−3ω1+3ω2⊕M−ω1+ω2⊕Mω1−ω2⊕M3ω1−3ω2⊕M−3ω1+ω2⊕M−ω1−ω2⊕Mω1−3ω2⊕M−3ω1−ω2⊕M−ω1−3ω2⊕M−3ω1−3ω2M6ω2⊕M4ω2⊕M2ω2⊕M0⊕M−2ω2⊕M−4ω2⊕M−6ω2

Semisimple subalgebra: W_{1}+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, 1.00)
0: (1.00, 0.00): (750.00, 300.00)
1: (0.00, 1.00): (200.00, 800.00)




Made total 4368 arithmetic operations while solving the Serre relations polynomial system.