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Subalgebra A12+2A11 ↪ A16
51 out of 61
Computations done by the calculator project.

Subalgebra type: A12+2A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A12+A11 .
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, 0, -1, 0, 0), (2, -2, -1, 0, -4, -2)

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, A11: (0, 1, 1, 1, 0, 0): 2, A11: (0, 0, 1, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: g−21, g6, g−13, g−3
Positive simple generators: g21, g−6, g13, g3
Cartan symmetric matrix: (2−100−120000200002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−100−120000200002)
Decomposition of ambient Lie algebra: V2ω4⊕2Vω3+ω4⊕Vω2+ω4⊕Vω1+ω4⊕V2ω3⊕Vω2+ω3⊕Vω1+ω3⊕Vω1+ω2⊕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+ω3−2ψ1+12ψ2⊕Vω3+ω4−4ψ1+10ψ2⊕Vω1+ω4+2ψ1+2ψ2⊕V2ω4⊕V2ω3⊕Vω1+ω2⊕2V0⊕Vω2+ω4−2ψ1−2ψ2⊕Vω3+ω4+4ψ1−10ψ2⊕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 11) ; the vectors are over the primal subalgebra.−h4+h2−h6−2h5−1/2h3−h2+h1g19g16g17g13g12g14g8g9g3
weight00ω1+ω2ω1+ω3ω2+ω32ω3ω1+ω4ω2+ω4ω3+ω4ω3+ω42ω4
weights rel. to Cartan of (centralizer+semisimple s.a.). 00ω1+ω2ω1+ω3−2ψ1+12ψ2ω2+ω3+2ψ1−12ψ22ω3ω1+ω4+2ψ1+2ψ2ω2+ω4−2ψ1−2ψ2ω3+ω4+4ψ1−10ψ2ω3+ω4−4ψ1+10ψ22ω4
Isotypic module decomposition over primal subalgebra (total 10 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0, 0, 0)Vω1+ω2 → (1, 1, 0, 0, 0, 0)Vω1+ω3−2ψ1+12ψ2 → (1, 0, 1, 0, -2, 12)Vω2+ω3+2ψ1−12ψ2 → (0, 1, 1, 0, 2, -12)V2ω3 → (0, 0, 2, 0, 0, 0)Vω1+ω4+2ψ1+2ψ2 → (1, 0, 0, 1, 2, 2)Vω2+ω4−2ψ1−2ψ2 → (0, 1, 0, 1, -2, -2)Vω3+ω4+4ψ1−10ψ2 → (0, 0, 1, 1, 4, -10)Vω3+ω4−4ψ1+10ψ2 → (0, 0, 1, 1, -4, 10)V2ω4 → (0, 0, 0, 2, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
−h4+h2
−h6−2h5−1/2h3−h2+h1
Semisimple subalgebra component.
−g19
−g−6
g21
−h6
−h6−h5−h4−h3−h2−h1
−g−21
2g6
−g−19
g16
g−11
−g1
g−5
−g−20
−g−17
g17
−g20
g5
g−1
−g11
g−16
Semisimple subalgebra component.
−g13
h4+h3+h2
2g−13
g12
g−15
−g7
g−10
−g−18
−g−14
g14
−g18
g10
g−7
−g15
g−12
g8
g−4
−g2
−g−9
g9
−g−2
g4
−g−8
Semisimple subalgebra component.
−g3
h3
2g−3
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1+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		ω2+ω3
ω1−ω2+ω3
ω2−ω3
−ω1+ω3
ω1−ω2−ω3
−ω1−ω3		2ω3
0
−2ω3		ω1+ω4
−ω1+ω2+ω4
ω1−ω4
−ω2+ω4
−ω1+ω2−ω4
−ω2−ω4		ω2+ω4
ω1−ω2+ω4
ω2−ω4
−ω1+ω4
ω1−ω2−ω4
−ω1−ω4		ω3+ω4
−ω3+ω4
ω3−ω4
−ω3−ω4		ω3+ω4
−ω3+ω4
ω3−ω4
−ω3−ω4		2ω4
0
−2ω4
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2		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		2ω3
0
−2ω3		ω1+ω4+2ψ1+2ψ2
−ω1+ω2+ω4+2ψ1+2ψ2
ω1−ω4+2ψ1+2ψ2
−ω2+ω4+2ψ1+2ψ2
−ω1+ω2−ω4+2ψ1+2ψ2
−ω2−ω4+2ψ1+2ψ2		ω2+ω4−2ψ1−2ψ2
ω1−ω2+ω4−2ψ1−2ψ2
ω2−ω4−2ψ1−2ψ2
−ω1+ω4−2ψ1−2ψ2
ω1−ω2−ω4−2ψ1−2ψ2
−ω1−ω4−2ψ1−2ψ2		ω3+ω4+4ψ1−10ψ2
−ω3+ω4+4ψ1−10ψ2
ω3−ω4+4ψ1−10ψ2
−ω3−ω4+4ψ1−10ψ2		ω3+ω4−4ψ1+10ψ2
−ω3+ω4−4ψ1+10ψ2
ω3−ω4−4ψ1+10ψ2
−ω3−ω4−4ψ1+10ψ2		2ω4
0
−2ω4
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0Mω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2Mω1+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+2ψ1−12ψ2M2ω3⊕M0⊕M−2ω3Mω1+ω4+2ψ1+2ψ2⊕M−ω1+ω2+ω4+2ψ1+2ψ2⊕M−ω2+ω4+2ψ1+2ψ2⊕Mω1−ω4+2ψ1+2ψ2⊕M−ω1+ω2−ω4+2ψ1+2ψ2⊕M−ω2−ω4+2ψ1+2ψ2Mω2+ω4−2ψ1−2ψ2⊕Mω1−ω2+ω4−2ψ1−2ψ2⊕M−ω1+ω4−2ψ1−2ψ2⊕Mω2−ω4−2ψ1−2ψ2⊕Mω1−ω2−ω4−2ψ1−2ψ2⊕M−ω1−ω4−2ψ1−2ψ2Mω3+ω4+4ψ1−10ψ2⊕M−ω3+ω4+4ψ1−10ψ2⊕Mω3−ω4+4ψ1−10ψ2⊕M−ω3−ω4+4ψ1−10ψ2Mω3+ω4−4ψ1+10ψ2⊕M−ω3+ω4−4ψ1+10ψ2⊕Mω3−ω4−4ψ1+10ψ2⊕M−ω3−ω4−4ψ1+10ψ2M2ω4⊕M0⊕M−2ω4
Isotypic character2M0Mω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2Mω1+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+2ψ1−12ψ2M2ω3⊕M0⊕M−2ω3Mω1+ω4+2ψ1+2ψ2⊕M−ω1+ω2+ω4+2ψ1+2ψ2⊕M−ω2+ω4+2ψ1+2ψ2⊕Mω1−ω4+2ψ1+2ψ2⊕M−ω1+ω2−ω4+2ψ1+2ψ2⊕M−ω2−ω4+2ψ1+2ψ2Mω2+ω4−2ψ1−2ψ2⊕Mω1−ω2+ω4−2ψ1−2ψ2⊕M−ω1+ω4−2ψ1−2ψ2⊕Mω2−ω4−2ψ1−2ψ2⊕Mω1−ω2−ω4−2ψ1−2ψ2⊕M−ω1−ω4−2ψ1−2ψ2Mω3+ω4+4ψ1−10ψ2⊕M−ω3+ω4+4ψ1−10ψ2⊕Mω3−ω4+4ψ1−10ψ2⊕M−ω3−ω4+4ψ1−10ψ2Mω3+ω4−4ψ1+10ψ2⊕M−ω3+ω4−4ψ1+10ψ2⊕Mω3−ω4−4ψ1+10ψ2⊕M−ω3−ω4−4ψ1+10ψ2M2ω4⊕M0⊕M−2ω4

Semisimple subalgebra: W_{2}+W_{5}+W_{10}
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)
(0.00, 1.00, 0.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00, 0.00): (266.67, 333.33)
1: (0.00, 1.00, 0.00, 0.00, 0.00, 0.00): (233.33, 366.67)
2: (0.00, 0.00, 1.00, 0.00, 0.00, 0.00): (200.00, 300.00)
3: (0.00, 0.00, 0.00, 1.00, 0.00, 0.00): (200.00, 300.00)
4: (0.00, 0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
5: (0.00, 0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 300.00)




Made total 535 arithmetic operations while solving the Serre relations polynomial system.