Subalgebra A52 ↪ A15
24 out of 37
Computations done by the calculator project.

Subalgebra type: A52 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A51 .
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: A52: (2, 3, 3, 3, 2): 10, (0, -1, 0, -2, -2): 10
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: g−6+g−11+g−12, 2g5+g4+g2
Positive simple generators: 2g12+g11+2g6, g−2+2g−4+g−5
Cartan symmetric matrix: (2/5−1/5−1/52/5)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (10−5−510)
Decomposition of ambient Lie algebra: V2ω1+2ω2⊕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 2) ; the vectors are over the primal subalgebra.g8+2g7+2g1g10
weightω1+ω22ω1+2ω2
Isotypic module decomposition over primal subalgebra (total 2 isotypic components).
Isotypical components + highest weightVω1+ω2 → (1, 1)V2ω1+2ω2 → (2, 2)
Module label W1W2
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
−1/2g8−g7−g1
−1/2g−2−g−4−1/2g−5
g12+1/2g11+g6
−h5−h4−1/2h2
−h5−3/2h4−3/2h3−3/2h2−h1
−1/2g−6−1/2g−11−1/2g−12
2g5+g4+g2
−1/2g−1−1/2g−7−1/2g−8
g10
g3
−g13
g−9
−g8+g7
−g8+g1
2g15
2g−4−g−5
g−2+g−4−g−5
2g12−2g11
2g12−g11−g6
g−14
−2h5+2h4
−2h5+h4+h2
6g14
−2h5−h4−h3+2h2+h1
2g−11−g−12
12g5−6g4
g−6+4g−11−3g−12
10g5−4g4−2g2
g−15
2g−7−4g−8
24g9
g−1+4g−7−10g−8
2g−13
−6g−3
2g−10
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		2ω1+2ω2
3ω2
3ω1
−2ω1+4ω2
ω1+ω2
ω1+ω2
4ω1−2ω2
−ω1+2ω2
−ω1+2ω2
2ω1−ω2
2ω1−ω2
−3ω1+3ω2
0
0
3ω1−3ω2
0
−2ω1+ω2
ω1−2ω2
−2ω1+ω2
ω1−2ω2
−4ω1+2ω2
−ω1−ω2
2ω1−4ω2
−ω1−ω2
−3ω1
−3ω2
−2ω1−2ω2
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		2ω1+2ω2
3ω2
3ω1
−2ω1+4ω2
ω1+ω2
ω1+ω2
4ω1−2ω2
−ω1+2ω2
−ω1+2ω2
2ω1−ω2
2ω1−ω2
−3ω1+3ω2
0
0
3ω1−3ω2
0
−2ω1+ω2
ω1−2ω2
−2ω1+ω2
ω1−2ω2
−4ω1+2ω2
−ω1−ω2
2ω1−4ω2
−ω1−ω2
−3ω1
−3ω2
−2ω1−2ω2
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−ω2M2ω1+2ω2⊕M3ω2⊕M3ω1⊕M−2ω1+4ω2⊕2Mω1+ω2⊕M4ω1−2ω2⊕2M−ω1+2ω2⊕2M2ω1−ω2⊕M−3ω1+3ω2⊕3M0⊕M3ω1−3ω2⊕2M−2ω1+ω2⊕2Mω1−2ω2⊕M−4ω1+2ω2⊕2M−ω1−ω2⊕M2ω1−4ω2⊕M−3ω1⊕M−3ω2⊕M−2ω1−2ω2
Isotypic characterMω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2M2ω1+2ω2⊕M3ω2⊕M3ω1⊕M−2ω1+4ω2⊕2Mω1+ω2⊕M4ω1−2ω2⊕2M−ω1+2ω2⊕2M2ω1−ω2⊕M−3ω1+3ω2⊕3M0⊕M3ω1−3ω2⊕2M−2ω1+ω2⊕2Mω1−2ω2⊕M−4ω1+2ω2⊕2M−ω1−ω2⊕M2ω1−4ω2⊕M−3ω1⊕M−3ω2⊕M−2ω1−2ω2

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.
Canvas not supported




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, 1.00)
0: (1.00, 0.00): (533.33, 639.17)
1: (0.00, 1.00): (366.67, 805.83)




Made total 22366 arithmetic operations while solving the Serre relations polynomial system.