Subalgebra A21+A11A16
16 out of 61
Computations done by the calculator project.

Subalgebra type: A21+A11 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A21 .
Centralizer: A21 + T2 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A12+A21
Basis of Cartan of centralizer: 3 vectors: (0, 0, 1, -1, 0, 0), (0, 1, 0, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: 2A21+A11 .

Elements Cartan subalgebra scaled to act by two by components: A21: (1, 2, 2, 2, 2, 1): 4, A11: (0, 0, 1, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g17+g21, g9
Positive simple generators: g21+g17, g9
Cartan symmetric matrix: (1002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4002)
Decomposition of ambient Lie algebra: V2ω24Vω1+ω24V2ω12Vω24Vω15V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+4ψ1+2ψ3Vω2+6ψ12ψ2Vω1+4ψ1+2ψ22ψ3V2ω12ψ2+4ψ3Vω1+ω22ψ1+2ψ2+2ψ3V2ψ2+4ψ3Vω1+ω2+2ψ14ψ2+2ψ3V2ω22V2ω1Vω1+ω22ψ1+4ψ22ψ33V0Vω1+ω2+2ψ12ψ22ψ3V2ω1+2ψ24ψ3V2ψ24ψ3Vω14ψ12ψ2+2ψ3Vω26ψ1+2ψ2Vω14ψ12ψ3
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 20) ; the vectors are over the primal subalgebra.g6+g1h4+h3h6+h1h5+h2g1+g6g15g10g8g12g4g3g20g21g17g19g18g13g14g16g9
weight00000ω1ω1ω1ω1ω2ω22ω12ω12ω12ω1ω1+ω2ω1+ω2ω1+ω2ω1+ω22ω2
weights rel. to Cartan of (centralizer+semisimple s.a.). 2ψ24ψ30002ψ2+4ψ3ω14ψ12ψ3ω14ψ12ψ2+2ψ3ω1+4ψ1+2ψ22ψ3ω1+4ψ1+2ψ3ω26ψ1+2ψ2ω2+6ψ12ψ22ω1+2ψ24ψ32ω12ω12ω12ψ2+4ψ3ω1+ω2+2ψ12ψ22ψ3ω1+ω22ψ1+4ψ22ψ3ω1+ω2+2ψ14ψ2+2ψ3ω1+ω22ψ1+2ψ2+2ψ32ω2
Isotypic module decomposition over primal subalgebra (total 18 isotypic components).
Isotypical components + highest weightV2ψ24ψ3 → (0, 0, 0, 2, -4)V0 → (0, 0, 0, 0, 0)V2ψ2+4ψ3 → (0, 0, 0, -2, 4)Vω14ψ12ψ3 → (1, 0, -4, 0, -2)Vω14ψ12ψ2+2ψ3 → (1, 0, -4, -2, 2)Vω1+4ψ1+2ψ22ψ3 → (1, 0, 4, 2, -2)Vω1+4ψ1+2ψ3 → (1, 0, 4, 0, 2)Vω26ψ1+2ψ2 → (0, 1, -6, 2, 0)Vω2+6ψ12ψ2 → (0, 1, 6, -2, 0)V2ω1+2ψ24ψ3 → (2, 0, 0, 2, -4)V2ω1 → (2, 0, 0, 0, 0)V2ω12ψ2+4ψ3 → (2, 0, 0, -2, 4)Vω1+ω2+2ψ12ψ22ψ3 → (1, 1, 2, -2, -2)Vω1+ω22ψ1+4ψ22ψ3 → (1, 1, -2, 4, -2)Vω1+ω2+2ψ14ψ2+2ψ3 → (1, 1, 2, -4, 2)Vω1+ω22ψ1+2ψ2+2ψ3 → (1, 1, -2, 2, 2)V2ω2 → (0, 2, 0, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13W14W15W16W17W18
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g6+g1
Cartan of centralizer component.
h4+h3
h6+h1
h5+h2
g1+g6
g15
g12
g10
g8
g8
g10
g12
g15
g4
g3
g3
g4
g20
g6g1
2g19
Semisimple subalgebra component.
g21g17
h6+2h5+2h4+2h3+2h2+h1
2g17+2g21
g21
h6h5h4h3h2h1
2g21
g19
g1+g6
2g20
g18
g7
g11
g16
g13
g5
g2
g14
g14
g2
g5
g13
g16
g11
g7
g18
Semisimple subalgebra component.
g9
h4+h3
2g9
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above000ω1
ω1
ω1
ω1
ω1
ω1
ω1
ω1
ω2
ω2
ω2
ω2
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
ω1+ω2
ω1+ω2
ω1ω2
ω1ω2
ω1+ω2
ω1+ω2
ω1ω2
ω1ω2
ω1+ω2
ω1+ω2
ω1ω2
ω1ω2
ω1+ω2
ω1+ω2
ω1ω2
ω1ω2
2ω2
0
2ω2
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer2ψ24ψ302ψ2+4ψ3ω14ψ12ψ3
ω14ψ12ψ3
ω14ψ12ψ2+2ψ3
ω14ψ12ψ2+2ψ3
ω1+4ψ1+2ψ22ψ3
ω1+4ψ1+2ψ22ψ3
ω1+4ψ1+2ψ3
ω1+4ψ1+2ψ3
ω26ψ1+2ψ2
ω26ψ1+2ψ2
ω2+6ψ12ψ2
ω2+6ψ12ψ2
2ω1+2ψ24ψ3
2ψ24ψ3
2ω1+2ψ24ψ3
2ω1
0
2ω1
2ω1
0
2ω1
2ω12ψ2+4ψ3
2ψ2+4ψ3
2ω12ψ2+4ψ3
ω1+ω2+2ψ12ψ22ψ3
ω1+ω2+2ψ12ψ22ψ3
ω1ω2+2ψ12ψ22ψ3
ω1ω2+2ψ12ψ22ψ3
ω1+ω22ψ1+4ψ22ψ3
ω1+ω22ψ1+4ψ22ψ3
ω1ω22ψ1+4ψ22ψ3
ω1ω22ψ1+4ψ22ψ3
ω1+ω2+2ψ14ψ2+2ψ3
ω1+ω2+2ψ14ψ2+2ψ3
ω1ω2+2ψ14ψ2+2ψ3
ω1ω2+2ψ14ψ2+2ψ3
ω1+ω22ψ1+2ψ2+2ψ3
ω1+ω22ψ1+2ψ2+2ψ3
ω1ω22ψ1+2ψ2+2ψ3
ω1ω22ψ1+2ψ2+2ψ3
2ω2
0
2ω2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M2ψ24ψ3M0M2ψ2+4ψ3Mω14ψ12ψ3Mω14ψ12ψ3Mω14ψ12ψ2+2ψ3Mω14ψ12ψ2+2ψ3Mω1+4ψ1+2ψ22ψ3Mω1+4ψ1+2ψ22ψ3Mω1+4ψ1+2ψ3Mω1+4ψ1+2ψ3Mω26ψ1+2ψ2Mω26ψ1+2ψ2Mω2+6ψ12ψ2Mω2+6ψ12ψ2M2ω1+2ψ24ψ3M2ψ24ψ3M2ω1+2ψ24ψ3M2ω1M0M2ω1M2ω1M0M2ω1M2ω12ψ2+4ψ3M2ψ2+4ψ3M2ω12ψ2+4ψ3Mω1+ω2+2ψ12ψ22ψ3Mω1+ω2+2ψ12ψ22ψ3Mω1ω2+2ψ12ψ22ψ3Mω1ω2+2ψ12ψ22ψ3Mω1+ω22ψ1+4ψ22ψ3Mω1+ω22ψ1+4ψ22ψ3Mω1ω22ψ1+4ψ22ψ3Mω1ω22ψ1+4ψ22ψ3Mω1+ω2+2ψ14ψ2+2ψ3Mω1+ω2+2ψ14ψ2+2ψ3Mω1ω2+2ψ14ψ2+2ψ3Mω1ω2+2ψ14ψ2+2ψ3Mω1+ω22ψ1+2ψ2+2ψ3Mω1+ω22ψ1+2ψ2+2ψ3Mω1ω22ψ1+2ψ2+2ψ3Mω1ω22ψ1+2ψ2+2ψ3M2ω2M0M2ω2
Isotypic characterM2ψ24ψ33M0M2ψ2+4ψ3Mω14ψ12ψ3Mω14ψ12ψ3Mω14ψ12ψ2+2ψ3Mω14ψ12ψ2+2ψ3Mω1+4ψ1+2ψ22ψ3Mω1+4ψ1+2ψ22ψ3Mω1+4ψ1+2ψ3Mω1+4ψ1+2ψ3Mω26ψ1+2ψ2Mω26ψ1+2ψ2Mω2+6ψ12ψ2Mω2+6ψ12ψ2M2ω1+2ψ24ψ3M2ψ24ψ3M2ω1+2ψ24ψ3M2ω1M0M2ω1M2ω1M0M2ω1M2ω12ψ2+4ψ3M2ψ2+4ψ3M2ω12ψ2+4ψ3Mω1+ω2+2ψ12ψ22ψ3Mω1+ω2+2ψ12ψ22ψ3Mω1ω2+2ψ12ψ22ψ3Mω1ω2+2ψ12ψ22ψ3Mω1+ω22ψ1+4ψ22ψ3Mω1+ω22ψ1+4ψ22ψ3Mω1ω22ψ1+4ψ22ψ3Mω1ω22ψ1+4ψ22ψ3Mω1+ω2+2ψ14ψ2+2ψ3Mω1+ω2+2ψ14ψ2+2ψ3Mω1ω2+2ψ14ψ2+2ψ3Mω1ω2+2ψ14ψ2+2ψ3Mω1+ω22ψ1+2ψ2+2ψ3Mω1+ω22ψ1+2ψ2+2ψ3Mω1ω22ψ1+2ψ2+2ψ3Mω1ω22ψ1+2ψ2+2ψ3M2ω2M0M2ω2

Semisimple subalgebra: W_{11}+W_{18}
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): (300.00, 300.00)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (200.00, 350.00)
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 622 arithmetic operations while solving the Serre relations polynomial system.