Subalgebra A12+A21 ↪ A16
40 out of 61
Computations done by the calculator project.

Subalgebra type: A12+A21 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A12 .
Centralizer: A21 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A12+A21
Basis of Cartan of centralizer: 2 vectors: (0, 1, 0, -1, 0, 0), (2, -2, -1, 0, -4, -2)
Contained up to conjugation as a direct summand of: A12+2A21 .

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, A21: (0, 1, 2, 1, 0, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: g−21, g6, g−3+g−13
Positive simple generators: g21, g−6, g13+g3
Cartan symmetric matrix: (2−10−120001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−10−120004)
Decomposition of ambient Lie algebra: 4V2ω3⊕2Vω2+ω3⊕2Vω1+ω3⊕Vω1+ω2⊕4V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+ω3−2ψ1+12ψ2⊕V2ω3−4ψ1+10ψ2⊕V−4ψ1+10ψ2⊕Vω1+ω3+2ψ1+2ψ2⊕2V2ω3⊕Vω1+ω2⊕2V0⊕Vω2+ω3−2ψ1−2ψ2⊕V2ω3+4ψ1−10ψ2⊕V4ψ1−10ψ2⊕Vω2+ω3+2ψ1−12ψ2
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 13) ; the vectors are over the primal subalgebra.g2+g−4−h6−2h5−1/2h3−h2+h1−h4+h2g4+g−2g19g12g16g17g14g8g3g13g9
weight0000ω1+ω2ω1+ω3ω1+ω3ω2+ω3ω2+ω32ω32ω32ω32ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). 4ψ1−10ψ200−4ψ1+10ψ2ω1+ω2ω1+ω3+2ψ1+2ψ2ω1+ω3−2ψ1+12ψ2ω2+ω3+2ψ1−12ψ2ω2+ω3−2ψ1−2ψ22ω3+4ψ1−10ψ22ω32ω32ω3−4ψ1+10ψ2
Isotypic module decomposition over primal subalgebra (total 12 isotypic components).
Isotypical components + highest weightV4ψ1−10ψ2 → (0, 0, 0, 4, -10)V0 → (0, 0, 0, 0, 0)V−4ψ1+10ψ2 → (0, 0, 0, -4, 10)Vω1+ω2 → (1, 1, 0, 0, 0)Vω1+ω3+2ψ1+2ψ2 → (1, 0, 1, 2, 2)Vω1+ω3−2ψ1+12ψ2 → (1, 0, 1, -2, 12)Vω2+ω3+2ψ1−12ψ2 → (0, 1, 1, 2, -12)Vω2+ω3−2ψ1−2ψ2 → (0, 1, 1, -2, -2)V2ω3+4ψ1−10ψ2 → (0, 0, 2, 4, -10)V2ω3 → (0, 0, 2, 0, 0)V2ω3−4ψ1+10ψ2 → (0, 0, 2, -4, 10)
Module label W1W2W3W4W5W6W7W8W9W10W11W12
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g2+g−4
Cartan of centralizer component.
−h6−2h5−1/2h3−h2+h1
−h4+h2
g4+g−2
Semisimple subalgebra component.
−g19
−g−6
g21
−h6
−h6−h5−h4−h3−h2−h1
−g−21
2g6
−g−19
g12
g−15
−g7
g−10
−g−18
−g−14
g16
g−11
−g1
g−5
−g−20
−g−17
g17
−g20
g5
g−1
−g11
g−16
g14
−g18
g10
g−7
−g15
g−12
g8
−g2+g−4
−2g−9
Semisimple subalgebra component.
g13+g3
−h4−2h3−h2
−2g−3−2g−13
g3
−h3
−2g−3
g9
g4−g−2
−2g−8
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above000ω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2		ω1+ω3
−ω1+ω2+ω3
ω1−ω3
−ω2+ω3
−ω1+ω2−ω3
−ω2−ω3		ω1+ω3
−ω1+ω2+ω3
ω1−ω3
−ω2+ω3
−ω1+ω2−ω3
−ω2−ω3		ω2+ω3
ω1−ω2+ω3
ω2−ω3
−ω1+ω3
ω1−ω2−ω3
−ω1−ω3		ω2+ω3
ω1−ω2+ω3
ω2−ω3
−ω1+ω3
ω1−ω2−ω3
−ω1−ω3		2ω3
0
−2ω3		2ω3
0
−2ω3		2ω3
0
−2ω3		2ω3
0
−2ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer4ψ1−10ψ20−4ψ1+10ψ2ω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2		ω1+ω3+2ψ1+2ψ2
−ω1+ω2+ω3+2ψ1+2ψ2
ω1−ω3+2ψ1+2ψ2
−ω2+ω3+2ψ1+2ψ2
−ω1+ω2−ω3+2ψ1+2ψ2
−ω2−ω3+2ψ1+2ψ2		ω1+ω3−2ψ1+12ψ2
−ω1+ω2+ω3−2ψ1+12ψ2
ω1−ω3−2ψ1+12ψ2
−ω2+ω3−2ψ1+12ψ2
−ω1+ω2−ω3−2ψ1+12ψ2
−ω2−ω3−2ψ1+12ψ2		ω2+ω3+2ψ1−12ψ2
ω1−ω2+ω3+2ψ1−12ψ2
ω2−ω3+2ψ1−12ψ2
−ω1+ω3+2ψ1−12ψ2
ω1−ω2−ω3+2ψ1−12ψ2
−ω1−ω3+2ψ1−12ψ2		ω2+ω3−2ψ1−2ψ2
ω1−ω2+ω3−2ψ1−2ψ2
ω2−ω3−2ψ1−2ψ2
−ω1+ω3−2ψ1−2ψ2
ω1−ω2−ω3−2ψ1−2ψ2
−ω1−ω3−2ψ1−2ψ2		2ω3+4ψ1−10ψ2
4ψ1−10ψ2
−2ω3+4ψ1−10ψ2		2ω3
0
−2ω3		2ω3
0
−2ω3		2ω3−4ψ1+10ψ2
−4ψ1+10ψ2
−2ω3−4ψ1+10ψ2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M4ψ1−10ψ2M0M−4ψ1+10ψ2Mω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2Mω1+ω3+2ψ1+2ψ2⊕M−ω1+ω2+ω3+2ψ1+2ψ2⊕M−ω2+ω3+2ψ1+2ψ2⊕Mω1−ω3+2ψ1+2ψ2⊕M−ω1+ω2−ω3+2ψ1+2ψ2⊕M−ω2−ω3+2ψ1+2ψ2Mω1+ω3−2ψ1+12ψ2⊕M−ω1+ω2+ω3−2ψ1+12ψ2⊕M−ω2+ω3−2ψ1+12ψ2⊕Mω1−ω3−2ψ1+12ψ2⊕M−ω1+ω2−ω3−2ψ1+12ψ2⊕M−ω2−ω3−2ψ1+12ψ2Mω2+ω3+2ψ1−12ψ2⊕Mω1−ω2+ω3+2ψ1−12ψ2⊕M−ω1+ω3+2ψ1−12ψ2⊕Mω2−ω3+2ψ1−12ψ2⊕Mω1−ω2−ω3+2ψ1−12ψ2⊕M−ω1−ω3+2ψ1−12ψ2Mω2+ω3−2ψ1−2ψ2⊕Mω1−ω2+ω3−2ψ1−2ψ2⊕M−ω1+ω3−2ψ1−2ψ2⊕Mω2−ω3−2ψ1−2ψ2⊕Mω1−ω2−ω3−2ψ1−2ψ2⊕M−ω1−ω3−2ψ1−2ψ2M2ω3+4ψ1−10ψ2⊕M4ψ1−10ψ2⊕M−2ω3+4ψ1−10ψ2M2ω3⊕M0⊕M−2ω3M2ω3⊕M0⊕M−2ω3M2ω3−4ψ1+10ψ2⊕M−4ψ1+10ψ2⊕M−2ω3−4ψ1+10ψ2
Isotypic characterM4ψ1−10ψ22M0M−4ψ1+10ψ2Mω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2Mω1+ω3+2ψ1+2ψ2⊕M−ω1+ω2+ω3+2ψ1+2ψ2⊕M−ω2+ω3+2ψ1+2ψ2⊕Mω1−ω3+2ψ1+2ψ2⊕M−ω1+ω2−ω3+2ψ1+2ψ2⊕M−ω2−ω3+2ψ1+2ψ2Mω1+ω3−2ψ1+12ψ2⊕M−ω1+ω2+ω3−2ψ1+12ψ2⊕M−ω2+ω3−2ψ1+12ψ2⊕Mω1−ω3−2ψ1+12ψ2⊕M−ω1+ω2−ω3−2ψ1+12ψ2⊕M−ω2−ω3−2ψ1+12ψ2Mω2+ω3+2ψ1−12ψ2⊕Mω1−ω2+ω3+2ψ1−12ψ2⊕M−ω1+ω3+2ψ1−12ψ2⊕Mω2−ω3+2ψ1−12ψ2⊕Mω1−ω2−ω3+2ψ1−12ψ2⊕M−ω1−ω3+2ψ1−12ψ2Mω2+ω3−2ψ1−2ψ2⊕Mω1−ω2+ω3−2ψ1−2ψ2⊕M−ω1+ω3−2ψ1−2ψ2⊕Mω2−ω3−2ψ1−2ψ2⊕Mω1−ω2−ω3−2ψ1−2ψ2⊕M−ω1−ω3−2ψ1−2ψ2M2ω3+4ψ1−10ψ2⊕M4ψ1−10ψ2⊕M−2ω3+4ψ1−10ψ2M2ω3⊕M0⊕M−2ω3M2ω3⊕M0⊕M−2ω3M2ω3−4ψ1+10ψ2⊕M−4ψ1+10ψ2⊕M−2ω3−4ψ1+10ψ2

Semisimple subalgebra: W_{4}+W_{10}
Centralizer extension: W_{1}+W_{2}+W_{3}

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 206642 arithmetic operations while solving the Serre relations polynomial system.