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Subalgebra B12+A11 ↪ E16
72 out of 119
Computations done by the calculator project.

Subalgebra type: B12+A11 (click on type for detailed printout).
Subalgebra is (parabolically) induced from B12 .
Centralizer: A11 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A15
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2)
Contained up to conjugation as a direct summand of: B12+2A11 .

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, A11: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: g−36, g14+g13, g−15
Positive simple generators: g36, g−13+g−14, g15
Cartan symmetric matrix: (2−10−110002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−20−240002)
Decomposition of ambient Lie algebra: V2ω3⊕2Vω2+ω3⊕2Vω1+ω3⊕V2ω2⊕2Vω3⊕4Vω2⊕Vω1⊕4V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω2+2ψ1+6ψ2⊕Vω2+ω3+6ψ2⊕Vω2−2ψ1+6ψ2⊕V4ψ1⊕Vω1+ω3+2ψ1⊕Vω3+2ψ1⊕V2ω3⊕V2ω2⊕Vω1⊕2V0⊕Vω1+ω3−2ψ1⊕Vω3−2ψ1⊕Vω2+2ψ1−6ψ2⊕V−4ψ1⊕Vω2+ω3−6ψ2⊕Vω2−2ψ1−6ψ2
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 17) ; the vectors are over the primal subalgebra.g−4−h6−1/2h5+1/2h3+h1h4g4g33+g32g11g16g7g12g5+g3−g10+g9g24g34g35g21g18g15
weight0000ω1ω2ω2ω2ω2ω3ω32ω2ω1+ω3ω1+ω3ω2+ω3ω2+ω32ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). −4ψ1004ψ1ω1ω2−2ψ1−6ψ2ω2+2ψ1−6ψ2ω2−2ψ1+6ψ2ω2+2ψ1+6ψ2ω3−2ψ1ω3+2ψ12ω2ω1+ω3−2ψ1ω1+ω3+2ψ1ω2+ω3−6ψ2ω2+ω3+6ψ22ω3
Isotypic module decomposition over primal subalgebra (total 16 isotypic components).
Isotypical components + highest weightV−4ψ1 → (0, 0, 0, -4, 0)V0 → (0, 0, 0, 0, 0)V4ψ1 → (0, 0, 0, 4, 0)Vω1 → (1, 0, 0, 0, 0)Vω2−2ψ1−6ψ2 → (0, 1, 0, -2, -6)Vω2+2ψ1−6ψ2 → (0, 1, 0, 2, -6)Vω2−2ψ1+6ψ2 → (0, 1, 0, -2, 6)Vω2+2ψ1+6ψ2 → (0, 1, 0, 2, 6)Vω3−2ψ1 → (0, 0, 1, -2, 0)Vω3+2ψ1 → (0, 0, 1, 2, 0)V2ω2 → (0, 2, 0, 0, 0)Vω1+ω3−2ψ1 → (1, 0, 1, -2, 0)Vω1+ω3+2ψ1 → (1, 0, 1, 2, 0)Vω2+ω3−6ψ2 → (0, 1, 1, 0, -6)Vω2+ω3+6ψ2 → (0, 1, 1, 0, 6)V2ω3 → (0, 0, 2, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13W14W15W16
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g−4
Cartan of centralizer component.
−h6−1/2h5+1/2h3+h1
h4
g4
g33+g32
g−13−g−14
−h5+h3
2g14−2g13
2g−32+2g−33
g11
g25
g−26
−g−12
g16
g28
−g−22
−g−7
g7
−g22
−g−28
g−16
g12
g26
−g−25
g−11
g5+g3
−g−9+g−10
−g10+g9
g−3+g−5
Semisimple subalgebra component.
−g24
−g33+g32
−g−13−g−14
2g36
−h5−2h4−h3−2h2
−2h6−4h5−6h4−4h3−4h2−2h1
−2g−36
2g14+2g13
2g−32−2g−33
4g−24
g34
−g−8
−g27
g5−g3
g−23
2g19
−g−9−g−10
−2g−30
−2g2
−2g−35
g35
−g−2
−g30
g10+g9
−g−19
2g23
−g−3+g−5
2g−27
2g8
2g−34
g21
g31
g6
−g−17
g20
−g−1
−g−29
−g−18
g18
−g29
−g1
−g−20
g17
g−6
g−31
−g−21
Semisimple subalgebra component.
−g15
h5+h4+h3
2g−15
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above000ω1
−ω1+2ω2
0
ω1−2ω2
−ω1		ω2
ω1−ω2
−ω1+ω2
−ω2		ω2
ω1−ω2
−ω1+ω2
−ω2		ω2
ω1−ω2
−ω1+ω2
−ω2		ω2
ω1−ω2
−ω1+ω2
−ω2		ω3
−ω3		ω3
−ω3		2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		ω1+ω3
−ω1+2ω2+ω3
ω1−ω3
ω3
−ω1+2ω2−ω3
ω1−2ω2+ω3
−ω3
−ω1+ω3
ω1−2ω2−ω3
−ω1−ω3		ω1+ω3
−ω1+2ω2+ω3
ω1−ω3
ω3
−ω1+2ω2−ω3
ω1−2ω2+ω3
−ω3
−ω1+ω3
ω1−2ω2−ω3
−ω1−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
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer−4ψ104ψ1ω1
−ω1+2ω2
0
ω1−2ω2
−ω1		ω2−2ψ1−6ψ2
ω1−ω2−2ψ1−6ψ2
−ω1+ω2−2ψ1−6ψ2
−ω2−2ψ1−6ψ2		ω2+2ψ1−6ψ2
ω1−ω2+2ψ1−6ψ2
−ω1+ω2+2ψ1−6ψ2
−ω2+2ψ1−6ψ2		ω2−2ψ1+6ψ2
ω1−ω2−2ψ1+6ψ2
−ω1+ω2−2ψ1+6ψ2
−ω2−2ψ1+6ψ2		ω2+2ψ1+6ψ2
ω1−ω2+2ψ1+6ψ2
−ω1+ω2+2ψ1+6ψ2
−ω2+2ψ1+6ψ2		ω3−2ψ1
−ω3−2ψ1		ω3+2ψ1
−ω3+2ψ1		2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		ω1+ω3−2ψ1
−ω1+2ω2+ω3−2ψ1
ω1−ω3−2ψ1
ω3−2ψ1
−ω1+2ω2−ω3−2ψ1
ω1−2ω2+ω3−2ψ1
−ω3−2ψ1
−ω1+ω3−2ψ1
ω1−2ω2−ω3−2ψ1
−ω1−ω3−2ψ1		ω1+ω3+2ψ1
−ω1+2ω2+ω3+2ψ1
ω1−ω3+2ψ1
ω3+2ψ1
−ω1+2ω2−ω3+2ψ1
ω1−2ω2+ω3+2ψ1
−ω3+2ψ1
−ω1+ω3+2ψ1
ω1−2ω2−ω3+2ψ1
−ω1−ω3+2ψ1		ω2+ω3−6ψ2
ω1−ω2+ω3−6ψ2
ω2−ω3−6ψ2
−ω1+ω2+ω3−6ψ2
ω1−ω2−ω3−6ψ2
−ω2+ω3−6ψ2
−ω1+ω2−ω3−6ψ2
−ω2−ω3−6ψ2		ω2+ω3+6ψ2
ω1−ω2+ω3+6ψ2
ω2−ω3+6ψ2
−ω1+ω2+ω3+6ψ2
ω1−ω2−ω3+6ψ2
−ω2+ω3+6ψ2
−ω1+ω2−ω3+6ψ2
−ω2−ω3+6ψ2		2ω3
0
−2ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M−4ψ1M0M4ψ1M−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2Mω2−2ψ1−6ψ2⊕M−ω1+ω2−2ψ1−6ψ2⊕Mω1−ω2−2ψ1−6ψ2⊕M−ω2−2ψ1−6ψ2Mω2+2ψ1−6ψ2⊕M−ω1+ω2+2ψ1−6ψ2⊕Mω1−ω2+2ψ1−6ψ2⊕M−ω2+2ψ1−6ψ2Mω2−2ψ1+6ψ2⊕M−ω1+ω2−2ψ1+6ψ2⊕Mω1−ω2−2ψ1+6ψ2⊕M−ω2−2ψ1+6ψ2Mω2+2ψ1+6ψ2⊕M−ω1+ω2+2ψ1+6ψ2⊕Mω1−ω2+2ψ1+6ψ2⊕M−ω2+2ψ1+6ψ2Mω3−2ψ1⊕M−ω3−2ψ1Mω3+2ψ1⊕M−ω3+2ψ1M2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M−ω1+2ω2+ω3−2ψ1⊕Mω1+ω3−2ψ1⊕Mω3−2ψ1⊕M−ω1+ω3−2ψ1⊕Mω1−2ω2+ω3−2ψ1⊕M−ω1+2ω2−ω3−2ψ1⊕Mω1−ω3−2ψ1⊕M−ω3−2ψ1⊕M−ω1−ω3−2ψ1⊕Mω1−2ω2−ω3−2ψ1M−ω1+2ω2+ω3+2ψ1⊕Mω1+ω3+2ψ1⊕Mω3+2ψ1⊕M−ω1+ω3+2ψ1⊕Mω1−2ω2+ω3+2ψ1⊕M−ω1+2ω2−ω3+2ψ1⊕Mω1−ω3+2ψ1⊕M−ω3+2ψ1⊕M−ω1−ω3+2ψ1⊕Mω1−2ω2−ω3+2ψ1Mω2+ω3−6ψ2⊕M−ω1+ω2+ω3−6ψ2⊕Mω1−ω2+ω3−6ψ2⊕M−ω2+ω3−6ψ2⊕Mω2−ω3−6ψ2⊕M−ω1+ω2−ω3−6ψ2⊕Mω1−ω2−ω3−6ψ2⊕M−ω2−ω3−6ψ2Mω2+ω3+6ψ2⊕M−ω1+ω2+ω3+6ψ2⊕Mω1−ω2+ω3+6ψ2⊕M−ω2+ω3+6ψ2⊕Mω2−ω3+6ψ2⊕M−ω1+ω2−ω3+6ψ2⊕Mω1−ω2−ω3+6ψ2⊕M−ω2−ω3+6ψ2M2ω3⊕M0⊕M−2ω3
Isotypic characterM−4ψ12M0M4ψ1M−ω1+2ω2⊕Mω1⊕M0⊕M−ω1⊕Mω1−2ω2Mω2−2ψ1−6ψ2⊕M−ω1+ω2−2ψ1−6ψ2⊕Mω1−ω2−2ψ1−6ψ2⊕M−ω2−2ψ1−6ψ2Mω2+2ψ1−6ψ2⊕M−ω1+ω2+2ψ1−6ψ2⊕Mω1−ω2+2ψ1−6ψ2⊕M−ω2+2ψ1−6ψ2Mω2−2ψ1+6ψ2⊕M−ω1+ω2−2ψ1+6ψ2⊕Mω1−ω2−2ψ1+6ψ2⊕M−ω2−2ψ1+6ψ2Mω2+2ψ1+6ψ2⊕M−ω1+ω2+2ψ1+6ψ2⊕Mω1−ω2+2ψ1+6ψ2⊕M−ω2+2ψ1+6ψ2Mω3−2ψ1⊕M−ω3−2ψ1Mω3+2ψ1⊕M−ω3+2ψ1M2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M−ω1+2ω2+ω3−2ψ1⊕Mω1+ω3−2ψ1⊕Mω3−2ψ1⊕M−ω1+ω3−2ψ1⊕Mω1−2ω2+ω3−2ψ1⊕M−ω1+2ω2−ω3−2ψ1⊕Mω1−ω3−2ψ1⊕M−ω3−2ψ1⊕M−ω1−ω3−2ψ1⊕Mω1−2ω2−ω3−2ψ1M−ω1+2ω2+ω3+2ψ1⊕Mω1+ω3+2ψ1⊕Mω3+2ψ1⊕M−ω1+ω3+2ψ1⊕Mω1−2ω2+ω3+2ψ1⊕M−ω1+2ω2−ω3+2ψ1⊕Mω1−ω3+2ψ1⊕M−ω3+2ψ1⊕M−ω1−ω3+2ψ1⊕Mω1−2ω2−ω3+2ψ1Mω2+ω3−6ψ2⊕M−ω1+ω2+ω3−6ψ2⊕Mω1−ω2+ω3−6ψ2⊕M−ω2+ω3−6ψ2⊕Mω2−ω3−6ψ2⊕M−ω1+ω2−ω3−6ψ2⊕Mω1−ω2−ω3−6ψ2⊕M−ω2−ω3−6ψ2Mω2+ω3+6ψ2⊕M−ω1+ω2+ω3+6ψ2⊕Mω1−ω2+ω3+6ψ2⊕M−ω2+ω3+6ψ2⊕Mω2−ω3+6ψ2⊕M−ω1+ω2−ω3+6ψ2⊕Mω1−ω2−ω3+6ψ2⊕M−ω2−ω3+6ψ2M2ω3⊕M0⊕M−2ω3

Semisimple subalgebra: W_{11}+W_{16}
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, 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 22148 arithmetic operations while solving the Serre relations polynomial system.