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Subalgebra A12+A41 ↪ A16
41 out of 61
Computations done by the calculator project.

Subalgebra type: A12+A41 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A12 .
Centralizer: T2 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A16
Basis of Cartan of centralizer: 2 vectors: (0, 1, 2, -1, 0, 0), (1, -1, -3, 0, -2, -1)

Elements Cartan subalgebra scaled to act by two by components: A12: (1, 1, 1, 1, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, A41: (0, 2, 2, 2, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: g−21, g6, g−2+g−9
Positive simple generators: g21, g−6, 2g9+2g2
Cartan symmetric matrix: (2−10−120001/2)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−10−120008)
Decomposition of ambient Lie algebra: V4ω3⊕Vω2+2ω3⊕Vω1+2ω3⊕3V2ω3⊕Vω1+ω2⊕Vω2⊕Vω1⊕2V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ω3−2ψ1+6ψ2⊕V2ω3−8ψ1+10ψ2⊕V4ω3⊕Vω1+6ψ1−4ψ2⊕V2ω3⊕Vω1+ω2⊕2V0⊕V2ω3+8ψ1−10ψ2⊕Vω2−6ψ1+4ψ2⊕Vω2+2ω3+2ψ1−6ψ2
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. As the centralizer is well-chosen and the centralizer of our subalgebra is non-trivial, we may in addition split highest weight vectors with the same weight over the semisimple part over the centralizer (recall that the centralizer preserves the weights over the subalgebra and in particular acts on the highest weight vectors). Therefore we have chosen our highest weight vectors to be, in addition, weight vectors over the Cartan of the centralizer of the starting subalgebra. Their weight over the sum of the Cartans of the semisimple subalgebra and its centralizer is indicated in the third row. The weights corresponding to the Cartan of the centralizer are again indicated with the letter \omega. As there is no preferred way of chosing a basis of the Cartan of the centralizer (unlike the starting semisimple Lie algebra: there we have a preferred basis induced by the fundamental weights), our centralizer weights are simply given by the constant by which the k^th basis element of the Cartan of the centralizer acts on the highest weight vector. Here, we use the choice for basis of the Cartan of the centralizer given at the start of the page.

Highest vectors of representations (total 11) ; the vectors are over the primal subalgebra.−h6−2h5−3h3−h2+h1−h4+2h3+h2g12g10g19g8g9+g2g4g16g17g13
weight00ω1ω2ω1+ω22ω32ω32ω3ω1+2ω3ω2+2ω34ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). 00ω1+6ψ1−4ψ2ω2−6ψ1+4ψ2ω1+ω22ω3+8ψ1−10ψ22ω32ω3−8ψ1+10ψ2ω1+2ω3−2ψ1+6ψ2ω2+2ω3+2ψ1−6ψ24ω3
Isotypic module decomposition over primal subalgebra (total 10 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0, 0)Vω1+6ψ1−4ψ2 → (1, 0, 0, 6, -4)Vω2−6ψ1+4ψ2 → (0, 1, 0, -6, 4)Vω1+ω2 → (1, 1, 0, 0, 0)V2ω3+8ψ1−10ψ2 → (0, 0, 2, 8, -10)V2ω3 → (0, 0, 2, 0, 0)V2ω3−8ψ1+10ψ2 → (0, 0, 2, -8, 10)Vω1+2ω3−2ψ1+6ψ2 → (1, 0, 2, -2, 6)Vω2+2ω3+2ψ1−6ψ2 → (0, 1, 2, 2, -6)V4ω3 → (0, 0, 4, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
−h6−2h5−3h3−h2+h1
−h4+2h3+h2
g12
g−15
g−10
g10
−g15
g−12
Semisimple subalgebra component.
−g19
−g−6
g21
−h6
−h6−h5−h4−h3−h2−h1
−g−21
2g6
−g−19
g8
g3
g−4
Semisimple subalgebra component.
−g9−g2
h4+h3+h2
g−2+g−9
g4
−g−3
g−8
g16
g−11
−g7
g−5
−g−18
g1
−g−14
g−20
g−17
g17
−g20
g14
g−1
−g18
g5
g−7
−g11
g−16
g13
g9−g2
−h4−h3+h2
3g−2−3g−9
6g−13
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1
−ω1+ω2
−ω2		ω2
ω1−ω2
−ω1		ω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2		2ω3
0
−2ω3		2ω3
0
−2ω3		2ω3
0
−2ω3		ω1+2ω3
−ω1+ω2+2ω3
ω1
−ω2+2ω3
−ω1+ω2
ω1−2ω3
−ω2
−ω1+ω2−2ω3
−ω2−2ω3		ω2+2ω3
ω1−ω2+2ω3
ω2
−ω1+2ω3
ω1−ω2
ω2−2ω3
−ω1
ω1−ω2−2ω3
−ω1−2ω3		4ω3
2ω3
0
−2ω3
−4ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1+6ψ1−4ψ2
−ω1+ω2+6ψ1−4ψ2
−ω2+6ψ1−4ψ2		ω2−6ψ1+4ψ2
ω1−ω2−6ψ1+4ψ2
−ω1−6ψ1+4ψ2		ω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2		2ω3+8ψ1−10ψ2
8ψ1−10ψ2
−2ω3+8ψ1−10ψ2		2ω3
0
−2ω3		2ω3−8ψ1+10ψ2
−8ψ1+10ψ2
−2ω3−8ψ1+10ψ2		ω1+2ω3−2ψ1+6ψ2
−ω1+ω2+2ω3−2ψ1+6ψ2
ω1−2ψ1+6ψ2
−ω2+2ω3−2ψ1+6ψ2
−ω1+ω2−2ψ1+6ψ2
ω1−2ω3−2ψ1+6ψ2
−ω2−2ψ1+6ψ2
−ω1+ω2−2ω3−2ψ1+6ψ2
−ω2−2ω3−2ψ1+6ψ2		ω2+2ω3+2ψ1−6ψ2
ω1−ω2+2ω3+2ψ1−6ψ2
ω2+2ψ1−6ψ2
−ω1+2ω3+2ψ1−6ψ2
ω1−ω2+2ψ1−6ψ2
ω2−2ω3+2ψ1−6ψ2
−ω1+2ψ1−6ψ2
ω1−ω2−2ω3+2ψ1−6ψ2
−ω1−2ω3+2ψ1−6ψ2		4ω3
2ω3
0
−2ω3
−4ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0Mω1+6ψ1−4ψ2⊕M−ω1+ω2+6ψ1−4ψ2⊕M−ω2+6ψ1−4ψ2Mω2−6ψ1+4ψ2⊕Mω1−ω2−6ψ1+4ψ2⊕M−ω1−6ψ1+4ψ2Mω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2M2ω3+8ψ1−10ψ2⊕M8ψ1−10ψ2⊕M−2ω3+8ψ1−10ψ2M2ω3⊕M0⊕M−2ω3M2ω3−8ψ1+10ψ2⊕M−8ψ1+10ψ2⊕M−2ω3−8ψ1+10ψ2Mω1+2ω3−2ψ1+6ψ2⊕M−ω1+ω2+2ω3−2ψ1+6ψ2⊕M−ω2+2ω3−2ψ1+6ψ2⊕Mω1−2ψ1+6ψ2⊕M−ω1+ω2−2ψ1+6ψ2⊕M−ω2−2ψ1+6ψ2⊕Mω1−2ω3−2ψ1+6ψ2⊕M−ω1+ω2−2ω3−2ψ1+6ψ2⊕M−ω2−2ω3−2ψ1+6ψ2Mω2+2ω3+2ψ1−6ψ2⊕Mω1−ω2+2ω3+2ψ1−6ψ2⊕M−ω1+2ω3+2ψ1−6ψ2⊕Mω2+2ψ1−6ψ2⊕Mω1−ω2+2ψ1−6ψ2⊕M−ω1+2ψ1−6ψ2⊕Mω2−2ω3+2ψ1−6ψ2⊕Mω1−ω2−2ω3+2ψ1−6ψ2⊕M−ω1−2ω3+2ψ1−6ψ2M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3
Isotypic character2M0Mω1+6ψ1−4ψ2⊕M−ω1+ω2+6ψ1−4ψ2⊕M−ω2+6ψ1−4ψ2Mω2−6ψ1+4ψ2⊕Mω1−ω2−6ψ1+4ψ2⊕M−ω1−6ψ1+4ψ2Mω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2M2ω3+8ψ1−10ψ2⊕M8ψ1−10ψ2⊕M−2ω3+8ψ1−10ψ2M2ω3⊕M0⊕M−2ω3M2ω3−8ψ1+10ψ2⊕M−8ψ1+10ψ2⊕M−2ω3−8ψ1+10ψ2Mω1+2ω3−2ψ1+6ψ2⊕M−ω1+ω2+2ω3−2ψ1+6ψ2⊕M−ω2+2ω3−2ψ1+6ψ2⊕Mω1−2ψ1+6ψ2⊕M−ω1+ω2−2ψ1+6ψ2⊕M−ω2−2ψ1+6ψ2⊕Mω1−2ω3−2ψ1+6ψ2⊕M−ω1+ω2−2ω3−2ψ1+6ψ2⊕M−ω2−2ω3−2ψ1+6ψ2Mω2+2ω3+2ψ1−6ψ2⊕Mω1−ω2+2ω3+2ψ1−6ψ2⊕M−ω1+2ω3+2ψ1−6ψ2⊕Mω2+2ψ1−6ψ2⊕Mω1−ω2+2ψ1−6ψ2⊕M−ω1+2ψ1−6ψ2⊕Mω2−2ω3+2ψ1−6ψ2⊕Mω1−ω2−2ω3+2ψ1−6ψ2⊕M−ω1−2ω3+2ψ1−6ψ2M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3

Semisimple subalgebra: W_{4}+W_{6}
Centralizer extension: W_{1}

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, 0.00)
(0.00, 1.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00): (266.67, 333.33)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (233.33, 366.67)
2: (0.00, 0.00, 1.00, 0.00, 0.00): (200.00, 300.00)
3: (0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
4: (0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 300.00)




Made total 267477 arithmetic operations while solving the Serre relations polynomial system.