Processing math: 100%
Subalgebra B12+A21 ↪ E16
73 out of 119
Computations done by the calculator project.

Subalgebra type: B12+A21 (click on type for detailed printout).
Subalgebra is (parabolically) induced from B12 .
Centralizer: T2 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 2 vectors: (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)

Elements Cartan subalgebra scaled to act by two by components: B12: (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, A21: (0, 0, 1, 2, 1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: g−36, g19+g8, −g−4+g−15
Positive simple generators: g36, g−8+g−19, g15−g4
Cartan symmetric matrix: (2−10−110001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−20−240004)
Decomposition of ambient Lie algebra: Vω1+2ω3⊕3V2ω3⊕4Vω2+ω3⊕V2ω2⊕2Vω1⊕2V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω2+ω3+2ψ1+2ψ2⊕Vω2+ω3−2ψ1+4ψ2⊕V2ω3+4ψ1−2ψ2⊕Vω1+2ω3⊕Vω1+4ψ1−2ψ2⊕V2ω3⊕V2ω2⊕V2ω3−4ψ1+2ψ2⊕2V0⊕Vω2+ω3+2ψ1−4ψ2⊕Vω1−4ψ1+2ψ2⊕Vω2+ω3−2ψ1−2ψ2
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.−h5+h3−h6+h1g33g32g24g16g21g18g12g10−g15+g4g9g35
weight00ω1ω12ω2ω2+ω3ω2+ω3ω2+ω3ω2+ω32ω32ω32ω3ω1+2ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). 00ω1−4ψ1+2ψ2ω1+4ψ1−2ψ22ω2ω2+ω3−2ψ1−2ψ2ω2+ω3+2ψ1−4ψ2ω2+ω3−2ψ1+4ψ2ω2+ω3+2ψ1+2ψ22ω3−4ψ1+2ψ22ω32ω3+4ψ1−2ψ2ω1+2ω3
Isotypic module decomposition over primal subalgebra (total 12 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0, 0)Vω1−4ψ1+2ψ2 → (1, 0, 0, -4, 2)Vω1+4ψ1−2ψ2 → (1, 0, 0, 4, -2)V2ω2 → (0, 2, 0, 0, 0)Vω2+ω3−2ψ1−2ψ2 → (0, 1, 1, -2, -2)Vω2+ω3+2ψ1−4ψ2 → (0, 1, 1, 2, -4)Vω2+ω3−2ψ1+4ψ2 → (0, 1, 1, -2, 4)Vω2+ω3+2ψ1+2ψ2 → (0, 1, 1, 2, 2)V2ω3−4ψ1+2ψ2 → (0, 0, 2, -4, 2)V2ω3 → (0, 0, 2, 0, 0)V2ω3+4ψ1−2ψ2 → (0, 0, 2, 4, -2)Vω1+2ω3 → (1, 0, 2, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
−h5+h3
−h6+h1
g33
g−13
−g5+g−3
−2g14
−2g−32
g32
−g−14
−g3+g−5
2g13
−2g−33
Semisimple subalgebra component.
−g24
−g34+g30
g−8+g−19
−2g36
h5+2h4+h3+2h2
2h6+4h5+6h4+4h3+4h2+2h1
2g−36
−2g19−2g8
2g−30−2g−34
4g−24
g16
−g31
−g11
g−17
−g20
g−7
g−29
−g−12
g21
−g28
g6
g−22
g25
g−1
g−26
g−18
g18
−g26
−g1
g−25
g22
−g−6
g−28
g−21
g12
g29
g7
g−20
−g17
−g−11
−g−31
−g−16
g10
−g5−g−3
2g−9
Semisimple subalgebra component.
g15−g4
−h5−2h4−h3
2g−4−2g−15
g9
g3+g−5
2g−10
g35
−g−2
−g34−g30
g15+g4
g−8−g−19
2g27
−2g23
−h5−h3
−2g−23
−2g−27
2g19−2g8
−2g−4−2g−15
−2g−30−2g−34
−4g2
−4g−35
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1
−ω1+2ω2
0
ω1−2ω2
−ω1		ω1
−ω1+2ω2
0
ω1−2ω2
−ω1		2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		ω2+ω3
ω1−ω2+ω3
ω2−ω3
−ω1+ω2+ω3
ω1−ω2−ω3
−ω2+ω3
−ω1+ω2−ω3
−ω2−ω3		ω2+ω3
ω1−ω2+ω3
ω2−ω3
−ω1+ω2+ω3
ω1−ω2−ω3
−ω2+ω3
−ω1+ω2−ω3
−ω2−ω3		ω2+ω3
ω1−ω2+ω3
ω2−ω3
−ω1+ω2+ω3
ω1−ω2−ω3
−ω2+ω3
−ω1+ω2−ω3
−ω2−ω3		ω2+ω3
ω1−ω2+ω3
ω2−ω3
−ω1+ω2+ω3
ω1−ω2−ω3
−ω2+ω3
−ω1+ω2−ω3
−ω2−ω3		2ω3
0
−2ω3		2ω3
0
−2ω3		2ω3
0
−2ω3		ω1+2ω3
−ω1+2ω2+2ω3
ω1
2ω3
−ω1+2ω2
ω1−2ω3
ω1−2ω2+2ω3
0
−ω1+2ω2−2ω3
−ω1+2ω3
ω1−2ω2
−2ω3
−ω1
ω1−2ω2−2ω3
−ω1−2ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1−4ψ1+2ψ2
−ω1+2ω2−4ψ1+2ψ2
−4ψ1+2ψ2
ω1−2ω2−4ψ1+2ψ2
−ω1−4ψ1+2ψ2		ω1+4ψ1−2ψ2
−ω1+2ω2+4ψ1−2ψ2
4ψ1−2ψ2
ω1−2ω2+4ψ1−2ψ2
−ω1+4ψ1−2ψ2		2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		ω2+ω3−2ψ1−2ψ2
ω1−ω2+ω3−2ψ1−2ψ2
ω2−ω3−2ψ1−2ψ2
−ω1+ω2+ω3−2ψ1−2ψ2
ω1−ω2−ω3−2ψ1−2ψ2
−ω2+ω3−2ψ1−2ψ2
−ω1+ω2−ω3−2ψ1−2ψ2
−ω2−ω3−2ψ1−2ψ2		ω2+ω3+2ψ1−4ψ2
ω1−ω2+ω3+2ψ1−4ψ2
ω2−ω3+2ψ1−4ψ2
−ω1+ω2+ω3+2ψ1−4ψ2
ω1−ω2−ω3+2ψ1−4ψ2
−ω2+ω3+2ψ1−4ψ2
−ω1+ω2−ω3+2ψ1−4ψ2
−ω2−ω3+2ψ1−4ψ2		ω2+ω3−2ψ1+4ψ2
ω1−ω2+ω3−2ψ1+4ψ2
ω2−ω3−2ψ1+4ψ2
−ω1+ω2+ω3−2ψ1+4ψ2
ω1−ω2−ω3−2ψ1+4ψ2
−ω2+ω3−2ψ1+4ψ2
−ω1+ω2−ω3−2ψ1+4ψ2
−ω2−ω3−2ψ1+4ψ2		ω2+ω3+2ψ1+2ψ2
ω1−ω2+ω3+2ψ1+2ψ2
ω2−ω3+2ψ1+2ψ2
−ω1+ω2+ω3+2ψ1+2ψ2
ω1−ω2−ω3+2ψ1+2ψ2
−ω2+ω3+2ψ1+2ψ2
−ω1+ω2−ω3+2ψ1+2ψ2
−ω2−ω3+2ψ1+2ψ2		2ω3−4ψ1+2ψ2
−4ψ1+2ψ2
−2ω3−4ψ1+2ψ2		2ω3
0
−2ω3		2ω3+4ψ1−2ψ2
4ψ1−2ψ2
−2ω3+4ψ1−2ψ2		ω1+2ω3
−ω1+2ω2+2ω3
ω1
2ω3
−ω1+2ω2
ω1−2ω3
ω1−2ω2+2ω3
0
−ω1+2ω2−2ω3
−ω1+2ω3
ω1−2ω2
−2ω3
−ω1
ω1−2ω2−2ω3
−ω1−2ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0M−ω1+2ω2−4ψ1+2ψ2⊕Mω1−4ψ1+2ψ2⊕M−4ψ1+2ψ2⊕M−ω1−4ψ1+2ψ2⊕Mω1−2ω2−4ψ1+2ψ2M−ω1+2ω2+4ψ1−2ψ2⊕Mω1+4ψ1−2ψ2⊕M4ψ1−2ψ2⊕M−ω1+4ψ1−2ψ2⊕Mω1−2ω2+4ψ1−2ψ2M2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2Mω2+ω3−2ψ1−2ψ2⊕M−ω1+ω2+ω3−2ψ1−2ψ2⊕Mω1−ω2+ω3−2ψ1−2ψ2⊕M−ω2+ω3−2ψ1−2ψ2⊕Mω2−ω3−2ψ1−2ψ2⊕M−ω1+ω2−ω3−2ψ1−2ψ2⊕Mω1−ω2−ω3−2ψ1−2ψ2⊕M−ω2−ω3−2ψ1−2ψ2Mω2+ω3+2ψ1−4ψ2⊕M−ω1+ω2+ω3+2ψ1−4ψ2⊕Mω1−ω2+ω3+2ψ1−4ψ2⊕M−ω2+ω3+2ψ1−4ψ2⊕Mω2−ω3+2ψ1−4ψ2⊕M−ω1+ω2−ω3+2ψ1−4ψ2⊕Mω1−ω2−ω3+2ψ1−4ψ2⊕M−ω2−ω3+2ψ1−4ψ2Mω2+ω3−2ψ1+4ψ2⊕M−ω1+ω2+ω3−2ψ1+4ψ2⊕Mω1−ω2+ω3−2ψ1+4ψ2⊕M−ω2+ω3−2ψ1+4ψ2⊕Mω2−ω3−2ψ1+4ψ2⊕M−ω1+ω2−ω3−2ψ1+4ψ2⊕Mω1−ω2−ω3−2ψ1+4ψ2⊕M−ω2−ω3−2ψ1+4ψ2Mω2+ω3+2ψ1+2ψ2⊕M−ω1+ω2+ω3+2ψ1+2ψ2⊕Mω1−ω2+ω3+2ψ1+2ψ2⊕M−ω2+ω3+2ψ1+2ψ2⊕Mω2−ω3+2ψ1+2ψ2⊕M−ω1+ω2−ω3+2ψ1+2ψ2⊕Mω1−ω2−ω3+2ψ1+2ψ2⊕M−ω2−ω3+2ψ1+2ψ2M2ω3−4ψ1+2ψ2⊕M−4ψ1+2ψ2⊕M−2ω3−4ψ1+2ψ2M2ω3⊕M0⊕M−2ω3M2ω3+4ψ1−2ψ2⊕M4ψ1−2ψ2⊕M−2ω3+4ψ1−2ψ2M−ω1+2ω2+2ω3⊕Mω1+2ω3⊕M2ω3⊕M−ω1+2ω3⊕Mω1−2ω2+2ω3⊕M−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2⊕M−ω1+2ω2−2ω3⊕Mω1−2ω3⊕M−2ω3⊕M−ω1−2ω3⊕Mω1−2ω2−2ω3
Isotypic character2M0M−ω1+2ω2−4ψ1+2ψ2⊕Mω1−4ψ1+2ψ2⊕M−4ψ1+2ψ2⊕M−ω1−4ψ1+2ψ2⊕Mω1−2ω2−4ψ1+2ψ2M−ω1+2ω2+4ψ1−2ψ2⊕Mω1+4ψ1−2ψ2⊕M4ψ1−2ψ2⊕M−ω1+4ψ1−2ψ2⊕Mω1−2ω2+4ψ1−2ψ2M2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2Mω2+ω3−2ψ1−2ψ2⊕M−ω1+ω2+ω3−2ψ1−2ψ2⊕Mω1−ω2+ω3−2ψ1−2ψ2⊕M−ω2+ω3−2ψ1−2ψ2⊕Mω2−ω3−2ψ1−2ψ2⊕M−ω1+ω2−ω3−2ψ1−2ψ2⊕Mω1−ω2−ω3−2ψ1−2ψ2⊕M−ω2−ω3−2ψ1−2ψ2Mω2+ω3+2ψ1−4ψ2⊕M−ω1+ω2+ω3+2ψ1−4ψ2⊕Mω1−ω2+ω3+2ψ1−4ψ2⊕M−ω2+ω3+2ψ1−4ψ2⊕Mω2−ω3+2ψ1−4ψ2⊕M−ω1+ω2−ω3+2ψ1−4ψ2⊕Mω1−ω2−ω3+2ψ1−4ψ2⊕M−ω2−ω3+2ψ1−4ψ2Mω2+ω3−2ψ1+4ψ2⊕M−ω1+ω2+ω3−2ψ1+4ψ2⊕Mω1−ω2+ω3−2ψ1+4ψ2⊕M−ω2+ω3−2ψ1+4ψ2⊕Mω2−ω3−2ψ1+4ψ2⊕M−ω1+ω2−ω3−2ψ1+4ψ2⊕Mω1−ω2−ω3−2ψ1+4ψ2⊕M−ω2−ω3−2ψ1+4ψ2Mω2+ω3+2ψ1+2ψ2⊕M−ω1+ω2+ω3+2ψ1+2ψ2⊕Mω1−ω2+ω3+2ψ1+2ψ2⊕M−ω2+ω3+2ψ1+2ψ2⊕Mω2−ω3+2ψ1+2ψ2⊕M−ω1+ω2−ω3+2ψ1+2ψ2⊕Mω1−ω2−ω3+2ψ1+2ψ2⊕M−ω2−ω3+2ψ1+2ψ2M2ω3−4ψ1+2ψ2⊕M−4ψ1+2ψ2⊕M−2ω3−4ψ1+2ψ2M2ω3⊕M0⊕M−2ω3M2ω3+4ψ1−2ψ2⊕M4ψ1−2ψ2⊕M−2ω3+4ψ1−2ψ2M−ω1+2ω2+2ω3⊕Mω1+2ω3⊕M2ω3⊕M−ω1+2ω3⊕Mω1−2ω2+2ω3⊕M−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2⊕M−ω1+2ω2−2ω3⊕Mω1−2ω3⊕M−2ω3⊕M−ω1−2ω3⊕Mω1−2ω2−2ω3

Semisimple subalgebra: W_{4}+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.
Canvas not supported




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, 350.00)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (250.00, 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 262431 arithmetic operations while solving the Serre relations polynomial system.