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Subalgebra D14 ↪ D14
23 out of 23
Computations done by the calculator project.

Subalgebra type: D14 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A13 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: D14

Elements Cartan subalgebra scaled to act by two by components: D14: (1, 2, 1, 1): 2, (0, -1, 0, 0): 2, (0, 0, 0, -1): 2, (0, 0, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 28.
Negative simple generators: g−12, g2, g4, g3
Positive simple generators: g12, g−2, g−4, g−3
Cartan symmetric matrix: (2−100−12−1−10−1200−102)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−100−12−1−10−1200−102)
Decomposition of ambient Lie algebra: Vω2
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
weightω2
Isotypic module decomposition over primal subalgebra (total 1 isotypic components).
Isotypical components + highest weightVω2 → (0, 1, 0, 0)
Module label W1
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
−g1
g5
−g−10
−g9
−g8
g−6
g−7
−g11
−g−3
g−2
−g−4
−g12
−h3
h2
−h4
h4+h3+2h2+h1
−g2
2g3
g−12
g4
−g−11
g7
g6
g−8
g−9
g10
g−5
g−1
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as aboveω2
ω1−ω2+ω3+ω4
−ω1+ω3+ω4
ω1−ω3+ω4
ω1+ω3−ω4
−ω1+ω2−ω3+ω4
−ω1+ω2+ω3−ω4
ω1+ω2−ω3−ω4
−ω2+2ω4
−ω1+2ω2−ω3−ω4
−ω2+2ω3
2ω1−ω2
0
0
0
0
ω1−2ω2+ω3+ω4
ω2−2ω4
−2ω1+ω2
ω2−2ω3
−ω1−ω2+ω3+ω4
ω1−ω2−ω3+ω4
ω1−ω2+ω3−ω4
−ω1−ω3+ω4
−ω1+ω3−ω4
ω1−ω3−ω4
−ω1+ω2−ω3−ω4
−ω2
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizerω2
ω1−ω2+ω3+ω4
−ω1+ω3+ω4
ω1−ω3+ω4
ω1+ω3−ω4
−ω1+ω2−ω3+ω4
−ω1+ω2+ω3−ω4
ω1+ω2−ω3−ω4
−ω2+2ω4
−ω1+2ω2−ω3−ω4
−ω2+2ω3
2ω1−ω2
0
0
0
0
ω1−2ω2+ω3+ω4
ω2−2ω4
−2ω1+ω2
ω2−2ω3
−ω1−ω2+ω3+ω4
ω1−ω2−ω3+ω4
ω1−ω2+ω3−ω4
−ω1−ω3+ω4
−ω1+ω3−ω4
ω1−ω3−ω4
−ω1+ω2−ω3−ω4
−ω2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.Mω1−ω2+ω3+ω4⊕M−ω2+2ω4⊕M−ω1+ω3+ω4⊕Mω1−2ω2+ω3+ω4⊕Mω1−ω3+ω4⊕M−ω2+2ω3⊕Mω2⊕M2ω1−ω2⊕Mω1+ω3−ω4⊕M−ω1−ω2+ω3+ω4⊕M−ω1+ω2−ω3+ω4⊕Mω1−ω2−ω3+ω4⊕4M0⊕M−ω1+ω2+ω3−ω4⊕Mω1−ω2+ω3−ω4⊕Mω1+ω2−ω3−ω4⊕M−ω1−ω3+ω4⊕M−2ω1+ω2⊕M−ω2⊕Mω2−2ω3⊕M−ω1+ω3−ω4⊕M−ω1+2ω2−ω3−ω4⊕Mω1−ω3−ω4⊕Mω2−2ω4⊕M−ω1+ω2−ω3−ω4
Isotypic characterMω1−ω2+ω3+ω4⊕M−ω2+2ω4⊕M−ω1+ω3+ω4⊕Mω1−2ω2+ω3+ω4⊕Mω1−ω3+ω4⊕M−ω2+2ω3⊕Mω2⊕M2ω1−ω2⊕Mω1+ω3−ω4⊕M−ω1−ω2+ω3+ω4⊕M−ω1+ω2−ω3+ω4⊕Mω1−ω2−ω3+ω4⊕4M0⊕M−ω1+ω2+ω3−ω4⊕Mω1−ω2+ω3−ω4⊕Mω1+ω2−ω3−ω4⊕M−ω1−ω3+ω4⊕M−2ω1+ω2⊕M−ω2⊕Mω2−2ω3⊕M−ω1+ω3−ω4⊕M−ω1+2ω2−ω3−ω4⊕Mω1−ω3−ω4⊕Mω2−2ω4⊕M−ω1+ω2−ω3−ω4

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, 490.00)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00, 0.00)
(0.00, 1.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00): (300.00, 590.00)
1: (0.00, 1.00, 0.00, 0.00): (300.00, 690.00)
2: (0.00, 0.00, 1.00, 0.00): (250.00, 590.00)
3: (0.00, 0.00, 0.00, 1.00): (250.00, 590.00)




Made total 527 arithmetic operations while solving the Serre relations polynomial system.