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Subalgebra A15 ↪ A15
37 out of 37
Computations done by the calculator project.

Subalgebra type: A15 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A14 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: A15

Elements Cartan subalgebra scaled to act by two by components: A15: (1, 1, 1, 1, 1): 2, (0, 0, 0, 0, -1): 2, (0, 0, 0, -1, 0): 2, (0, 0, -1, 0, 0): 2, (0, -1, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 35.
Negative simple generators: g−15, g5, g4, g3, g2
Positive simple generators: g15, g−5, g−4, g−3, g−2
Cartan symmetric matrix: (2−1000−12−1000−12−1000−12−1000−12)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−1000−12−1000−12−1000−12−1000−12)
Decomposition of ambient Lie algebra: Vω1+ω5
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 1) ; the vectors are over the primal subalgebra.g1
weightω1+ω5
Isotypic module decomposition over primal subalgebra (total 1 isotypic components).
Isotypical components + highest weightVω1+ω5 → (1, 0, 0, 0, 1)
Module label W1
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
−g1
−g−14
g6
−g−11
g−12
−g10
−g−7
g−8
−g−9
g13
−g−2
g−3
−g−4
g−5
−g15
−h2
h3
−h4
h5
h5+h4+h3+h2+h1
−g3
2g2
g4
−g5
g−15
g8
−g7
−g9
g−13
−g12
g11
g−10
g−6
−g14
g−1
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as aboveω1+ω5
−ω1+ω2+ω5
ω1+ω4−ω5
−ω2+ω3+ω5
−ω1+ω2+ω4−ω5
ω1+ω3−ω4
−ω3+ω4+ω5
−ω2+ω3+ω4−ω5
−ω1+ω2+ω3−ω4
ω1+ω2−ω3
−ω4+2ω5
−ω3+2ω4−ω5
−ω2+2ω3−ω4
−ω1+2ω2−ω3
2ω1−ω2
0
0
0
0
0
ω3−2ω4+ω5
ω4−2ω5
ω2−2ω3+ω4
ω1−2ω2+ω3
−2ω1+ω2
ω2−ω3−ω4+ω5
ω3−ω4−ω5
ω1−ω2−ω3+ω4
−ω1−ω2+ω3
ω1−ω2−ω4+ω5
ω2−ω3−ω5
−ω1−ω3+ω4
−ω1−ω4+ω5
ω1−ω2−ω5
−ω1−ω5
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizerω1+ω5
−ω1+ω2+ω5
ω1+ω4−ω5
−ω2+ω3+ω5
−ω1+ω2+ω4−ω5
ω1+ω3−ω4
−ω3+ω4+ω5
−ω2+ω3+ω4−ω5
−ω1+ω2+ω3−ω4
ω1+ω2−ω3
−ω4+2ω5
−ω3+2ω4−ω5
−ω2+2ω3−ω4
−ω1+2ω2−ω3
2ω1−ω2
0
0
0
0
0
ω3−2ω4+ω5
ω4−2ω5
ω2−2ω3+ω4
ω1−2ω2+ω3
−2ω1+ω2
ω2−ω3−ω4+ω5
ω3−ω4−ω5
ω1−ω2−ω3+ω4
−ω1−ω2+ω3
ω1−ω2−ω4+ω5
ω2−ω3−ω5
−ω1−ω3+ω4
−ω1−ω4+ω5
ω1−ω2−ω5
−ω1−ω5
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.Mω1+ω5⊕M−ω4+2ω5⊕M−ω3+ω4+ω5⊕M−ω2+ω3+ω5⊕M−ω1+ω2+ω5⊕M2ω1−ω2⊕Mω1+ω2−ω3⊕Mω1+ω3−ω4⊕Mω1+ω4−ω5⊕Mω1−ω2−ω4+ω5⊕Mω2−ω3−ω4+ω5⊕Mω3−2ω4+ω5⊕Mω1−ω2−ω3+ω4⊕Mω2−2ω3+ω4⊕Mω1−2ω2+ω3⊕5M0⊕M−ω1+2ω2−ω3⊕M−ω2+2ω3−ω4⊕M−ω1+ω2+ω3−ω4⊕M−ω3+2ω4−ω5⊕M−ω2+ω3+ω4−ω5⊕M−ω1+ω2+ω4−ω5⊕M−ω1−ω4+ω5⊕M−ω1−ω3+ω4⊕M−ω1−ω2+ω3⊕M−2ω1+ω2⊕Mω1−ω2−ω5⊕Mω2−ω3−ω5⊕Mω3−ω4−ω5⊕Mω4−2ω5⊕M−ω1−ω5
Isotypic characterMω1+ω5⊕M−ω4+2ω5⊕M−ω3+ω4+ω5⊕M−ω2+ω3+ω5⊕M−ω1+ω2+ω5⊕M2ω1−ω2⊕Mω1+ω2−ω3⊕Mω1+ω3−ω4⊕Mω1+ω4−ω5⊕Mω1−ω2−ω4+ω5⊕Mω2−ω3−ω4+ω5⊕Mω3−2ω4+ω5⊕Mω1−ω2−ω3+ω4⊕Mω2−2ω3+ω4⊕Mω1−2ω2+ω3⊕5M0⊕M−ω1+2ω2−ω3⊕M−ω2+2ω3−ω4⊕M−ω1+ω2+ω3−ω4⊕M−ω3+2ω4−ω5⊕M−ω2+ω3+ω4−ω5⊕M−ω1+ω2+ω4−ω5⊕M−ω1−ω4+ω5⊕M−ω1−ω3+ω4⊕M−ω1−ω2+ω3⊕M−2ω1+ω2⊕Mω1−ω2−ω5⊕Mω2−ω3−ω5⊕Mω3−ω4−ω5⊕Mω4−2ω5⊕M−ω1−ω5

Semisimple subalgebra: W_{1}
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, 612.50)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00, 0.00, 0.00)
(0.00, 1.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00): (283.33, 679.17)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (266.67, 745.83)
2: (0.00, 0.00, 1.00, 0.00, 0.00): (250.00, 712.50)
3: (0.00, 0.00, 0.00, 1.00, 0.00): (233.33, 679.17)
4: (0.00, 0.00, 0.00, 0.00, 1.00): (216.67, 645.83)




Made total 612 arithmetic operations while solving the Serre relations polynomial system.