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Subalgebra A12+2A41 ↪ E16
93 out of 119
Computations done by the calculator project.

Subalgebra type: A12+2A41 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A12+A41 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: E16

Elements Cartan subalgebra scaled to act by two by components: A12: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, A41: (0, 0, 0, 0, 2, 2): 8, A41: (2, 0, 2, 0, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: g−36, g2, g−5+g−6, g−1+g−3
Positive simple generators: g36, g−2, 2g6+2g5, 2g3+2g1
Cartan symmetric matrix: (2−100−1200001/200001/2)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−100−120000800008)
Decomposition of ambient Lie algebra: Vω2+2ω3+2ω4⊕Vω1+2ω3+2ω4⊕V4ω4⊕V4ω3⊕V2ω4⊕V2ω3⊕Vω1+ω2
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.
Highest vectors of representations (total 7) ; the vectors are over the primal subalgebra.g35g6+g5g3+g1g11g7g34g24
weightω1+ω22ω32ω44ω34ω4ω1+2ω3+2ω4ω2+2ω3+2ω4
Isotypic module decomposition over primal subalgebra (total 7 isotypic components).
Isotypical components + highest weightVω1+ω2 → (1, 1, 0, 0)V2ω3 → (0, 0, 2, 0)V2ω4 → (0, 0, 0, 2)V4ω3 → (0, 0, 4, 0)V4ω4 → (0, 0, 0, 4)Vω1+2ω3+2ω4 → (1, 0, 2, 2)Vω2+2ω3+2ω4 → (0, 1, 2, 2)
Module label W1W2W3W4W5W6W7
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
−g35
g−2
−g36
h2
h6+2h5+3h4+2h3+2h2+h1
g−36
−2g2
−g−35
Semisimple subalgebra component.
−g6−g5
h6+h5
g−5+g−6
Semisimple subalgebra component.
−g3−g1
h3+h1
g−1+g−3
g11
g6−g5
−h6+h5
3g−5−3g−6
6g−11
g7
g3−g1
−h3+h1
3g−1−3g−3
6g−7
g34
−g−8
g32
g33
g−4
−g−14
g−13
−g29
g30
g31
g−10
−g−9
−g−20
g−19
−g−17
−g26
g28
g−16
−g−15
g−12
g−25
−g−22
−g23
−g−21
g−18
−g−27
g−24
g24
g27
−g18
g21
−g−23
−g22
g25
g12
−g15
g16
−g−28
g−26
g17
−g19
g20
g9
−g10
g−31
g−30
−g−29
g13
−g14
g4
−g−33
−g−32
g8
g−34
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as aboveω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2		2ω3
0
−2ω3		2ω4
0
−2ω4		4ω3
2ω3
0
−2ω3
−4ω3		4ω4
2ω4
0
−2ω4
−4ω4		ω1+2ω3+2ω4
−ω1+ω2+2ω3+2ω4
ω1+2ω4
ω1+2ω3
−ω2+2ω3+2ω4
−ω1+ω2+2ω4
−ω1+ω2+2ω3
ω1−2ω3+2ω4
ω1
ω1+2ω3−2ω4
−ω2+2ω4
−ω2+2ω3
−ω1+ω2−2ω3+2ω4
−ω1+ω2
−ω1+ω2+2ω3−2ω4
ω1−2ω3
ω1−2ω4
−ω2−2ω3+2ω4
−ω2
−ω2+2ω3−2ω4
−ω1+ω2−2ω3
−ω1+ω2−2ω4
ω1−2ω3−2ω4
−ω2−2ω3
−ω2−2ω4
−ω1+ω2−2ω3−2ω4
−ω2−2ω3−2ω4		ω2+2ω3+2ω4
ω1−ω2+2ω3+2ω4
ω2+2ω4
ω2+2ω3
−ω1+2ω3+2ω4
ω1−ω2+2ω4
ω1−ω2+2ω3
ω2−2ω3+2ω4
ω2
ω2+2ω3−2ω4
−ω1+2ω4
−ω1+2ω3
ω1−ω2−2ω3+2ω4
ω1−ω2
ω1−ω2+2ω3−2ω4
ω2−2ω3
ω2−2ω4
−ω1−2ω3+2ω4
−ω1
−ω1+2ω3−2ω4
ω1−ω2−2ω3
ω1−ω2−2ω4
ω2−2ω3−2ω4
−ω1−2ω3
−ω1−2ω4
ω1−ω2−2ω3−2ω4
−ω1−2ω3−2ω4
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizerω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2		2ω3
0
−2ω3		2ω4
0
−2ω4		4ω3
2ω3
0
−2ω3
−4ω3		4ω4
2ω4
0
−2ω4
−4ω4		ω1+2ω3+2ω4
−ω1+ω2+2ω3+2ω4
ω1+2ω4
ω1+2ω3
−ω2+2ω3+2ω4
−ω1+ω2+2ω4
−ω1+ω2+2ω3
ω1−2ω3+2ω4
ω1
ω1+2ω3−2ω4
−ω2+2ω4
−ω2+2ω3
−ω1+ω2−2ω3+2ω4
−ω1+ω2
−ω1+ω2+2ω3−2ω4
ω1−2ω3
ω1−2ω4
−ω2−2ω3+2ω4
−ω2
−ω2+2ω3−2ω4
−ω1+ω2−2ω3
−ω1+ω2−2ω4
ω1−2ω3−2ω4
−ω2−2ω3
−ω2−2ω4
−ω1+ω2−2ω3−2ω4
−ω2−2ω3−2ω4		ω2+2ω3+2ω4
ω1−ω2+2ω3+2ω4
ω2+2ω4
ω2+2ω3
−ω1+2ω3+2ω4
ω1−ω2+2ω4
ω1−ω2+2ω3
ω2−2ω3+2ω4
ω2
ω2+2ω3−2ω4
−ω1+2ω4
−ω1+2ω3
ω1−ω2−2ω3+2ω4
ω1−ω2
ω1−ω2+2ω3−2ω4
ω2−2ω3
ω2−2ω4
−ω1−2ω3+2ω4
−ω1
−ω1+2ω3−2ω4
ω1−ω2−2ω3
ω1−ω2−2ω4
ω2−2ω3−2ω4
−ω1−2ω3
−ω1−2ω4
ω1−ω2−2ω3−2ω4
−ω1−2ω3−2ω4
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.Mω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2M2ω3⊕M0⊕M−2ω3M2ω4⊕M0⊕M−2ω4M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3M4ω4⊕M2ω4⊕M0⊕M−2ω4⊕M−4ω4Mω1+2ω3+2ω4⊕M−ω1+ω2+2ω3+2ω4⊕M−ω2+2ω3+2ω4⊕Mω1+2ω4⊕Mω1+2ω3⊕M−ω1+ω2+2ω4⊕M−ω1+ω2+2ω3⊕M−ω2+2ω4⊕Mω1−2ω3+2ω4⊕M−ω2+2ω3⊕Mω1⊕Mω1+2ω3−2ω4⊕M−ω1+ω2−2ω3+2ω4⊕M−ω1+ω2⊕M−ω1+ω2+2ω3−2ω4⊕M−ω2−2ω3+2ω4⊕M−ω2⊕Mω1−2ω3⊕M−ω2+2ω3−2ω4⊕Mω1−2ω4⊕M−ω1+ω2−2ω3⊕M−ω1+ω2−2ω4⊕M−ω2−2ω3⊕M−ω2−2ω4⊕Mω1−2ω3−2ω4⊕M−ω1+ω2−2ω3−2ω4⊕M−ω2−2ω3−2ω4Mω2+2ω3+2ω4⊕Mω1−ω2+2ω3+2ω4⊕M−ω1+2ω3+2ω4⊕Mω2+2ω4⊕Mω2+2ω3⊕Mω1−ω2+2ω4⊕Mω1−ω2+2ω3⊕M−ω1+2ω4⊕Mω2−2ω3+2ω4⊕M−ω1+2ω3⊕Mω2⊕Mω2+2ω3−2ω4⊕Mω1−ω2−2ω3+2ω4⊕Mω1−ω2⊕Mω1−ω2+2ω3−2ω4⊕M−ω1−2ω3+2ω4⊕M−ω1⊕Mω2−2ω3⊕M−ω1+2ω3−2ω4⊕Mω2−2ω4⊕Mω1−ω2−2ω3⊕Mω1−ω2−2ω4⊕M−ω1−2ω3⊕M−ω1−2ω4⊕Mω2−2ω3−2ω4⊕Mω1−ω2−2ω3−2ω4⊕M−ω1−2ω3−2ω4
Isotypic characterMω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2M2ω3⊕M0⊕M−2ω3M2ω4⊕M0⊕M−2ω4M4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3M4ω4⊕M2ω4⊕M0⊕M−2ω4⊕M−4ω4Mω1+2ω3+2ω4⊕M−ω1+ω2+2ω3+2ω4⊕M−ω2+2ω3+2ω4⊕Mω1+2ω4⊕Mω1+2ω3⊕M−ω1+ω2+2ω4⊕M−ω1+ω2+2ω3⊕M−ω2+2ω4⊕Mω1−2ω3+2ω4⊕M−ω2+2ω3⊕Mω1⊕Mω1+2ω3−2ω4⊕M−ω1+ω2−2ω3+2ω4⊕M−ω1+ω2⊕M−ω1+ω2+2ω3−2ω4⊕M−ω2−2ω3+2ω4⊕M−ω2⊕Mω1−2ω3⊕M−ω2+2ω3−2ω4⊕Mω1−2ω4⊕M−ω1+ω2−2ω3⊕M−ω1+ω2−2ω4⊕M−ω2−2ω3⊕M−ω2−2ω4⊕Mω1−2ω3−2ω4⊕M−ω1+ω2−2ω3−2ω4⊕M−ω2−2ω3−2ω4Mω2+2ω3+2ω4⊕Mω1−ω2+2ω3+2ω4⊕M−ω1+2ω3+2ω4⊕Mω2+2ω4⊕Mω2+2ω3⊕Mω1−ω2+2ω4⊕Mω1−ω2+2ω3⊕M−ω1+2ω4⊕Mω2−2ω3+2ω4⊕M−ω1+2ω3⊕Mω2⊕Mω2+2ω3−2ω4⊕Mω1−ω2−2ω3+2ω4⊕Mω1−ω2⊕Mω1−ω2+2ω3−2ω4⊕M−ω1−2ω3+2ω4⊕M−ω1⊕Mω2−2ω3⊕M−ω1+2ω3−2ω4⊕Mω2−2ω4⊕Mω1−ω2−2ω3⊕Mω1−ω2−2ω4⊕M−ω1−2ω3⊕M−ω1−2ω4⊕Mω2−2ω3−2ω4⊕Mω1−ω2−2ω3−2ω4⊕M−ω1−2ω3−2ω4

Semisimple subalgebra: W_{1}+W_{2}+W_{3}
Centralizer extension: 0

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, 472.50)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00, 0.00)
(0.00, 1.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00): (266.67, 505.83)
1: (0.00, 1.00, 0.00, 0.00): (233.33, 539.17)
2: (0.00, 0.00, 1.00, 0.00): (200.00, 472.50)
3: (0.00, 0.00, 0.00, 1.00): (200.00, 472.50)



Made total 3798 arithmetic operations while solving the Serre relations polynomial system.