Processing math: 100%
Subalgebra 3A11 ↪ E16
57 out of 119
Computations done by the calculator project.

Subalgebra type: 3A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from 2A11 .
Centralizer: A11 + T2 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A15
Basis of Cartan of centralizer: 3 vectors: (0, 0, 0, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: 4A11 .

Elements Cartan subalgebra scaled to act by two by components: A11: (1, 2, 2, 3, 2, 1): 2, A11: (1, 0, 1, 1, 1, 1): 2, A11: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: g−36, g−24, g−15
Positive simple generators: g36, g24, g15
Cartan symmetric matrix: (200020002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (200020002)
Decomposition of ambient Lie algebra: 2Vω1+ω2+ω3⊕V2ω3⊕2Vω2+ω3⊕2Vω1+ω3⊕V2ω2⊕2Vω1+ω2⊕V2ω1⊕4Vω3⊕4Vω2⊕4Vω1⊕5V0
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+2ψ2+2ψ3⊕Vω1+ω3+2ψ2+2ψ3⊕Vω1+2ψ1−2ψ2+4ψ3⊕Vω1+ω2+ω3+2ψ1⊕Vω3+2ψ1+4ψ2−2ψ3⊕Vω2+ω3−2ψ2+4ψ3⊕V4ψ1⊕Vω1+ω2+4ψ2−2ψ3⊕Vω2−2ψ1+2ψ2+2ψ3⊕V2ω3⊕V2ω2⊕V2ω1⊕Vω1−2ψ1−2ψ2+4ψ3⊕Vω3+2ψ1−4ψ2+2ψ3⊕Vω1+ω2+ω3−2ψ1⊕Vω3−2ψ1+4ψ2−2ψ3⊕Vω1+2ψ1+2ψ2−4ψ3⊕Vω1+ω2−4ψ2+2ψ3⊕3V0⊕Vω2+ω3+2ψ2−4ψ3⊕Vω2+2ψ1−2ψ2−2ψ3⊕Vω1+ω3−2ψ2−2ψ3⊕Vω3−2ψ1−4ψ2+2ψ3⊕Vω1−2ψ1+2ψ2−4ψ3⊕V−4ψ1⊕Vω2−2ψ1−2ψ2−2ψ3
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 28) ; the vectors are over the primal subalgebra.g−4−h5+h3−h6+h1h4g4g25g28g22g26g11g16g7g12g5g3g10g9g36g33g32g24g31g29g21g18g15g34g35
weight00000ω1ω1ω1ω1ω2ω2ω2ω2ω3ω3ω3ω32ω1ω1+ω2ω1+ω22ω2ω1+ω3ω1+ω3ω2+ω3ω2+ω32ω3ω1+ω2+ω3ω1+ω2+ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). −4ψ10004ψ1ω1−2ψ1+2ψ2−4ψ3ω1+2ψ1+2ψ2−4ψ3ω1−2ψ1−2ψ2+4ψ3ω1+2ψ1−2ψ2+4ψ3ω2−2ψ1−2ψ2−2ψ3ω2+2ψ1−2ψ2−2ψ3ω2−2ψ1+2ψ2+2ψ3ω2+2ψ1+2ψ2+2ψ3ω3−2ψ1−4ψ2+2ψ3ω3−2ψ1+4ψ2−2ψ3ω3+2ψ1−4ψ2+2ψ3ω3+2ψ1+4ψ2−2ψ32ω1ω1+ω2−4ψ2+2ψ3ω1+ω2+4ψ2−2ψ32ω2ω1+ω3−2ψ2−2ψ3ω1+ω3+2ψ2+2ψ3ω2+ω3+2ψ2−4ψ3ω2+ω3−2ψ2+4ψ32ω3ω1+ω2+ω3−2ψ1ω1+ω2+ω3+2ψ1
Isotypic module decomposition over primal subalgebra (total 26 isotypic components).
Isotypical components + highest weightV−4ψ1 → (0, 0, 0, -4, 0, 0)V0 → (0, 0, 0, 0, 0, 0)V4ψ1 → (0, 0, 0, 4, 0, 0)Vω1−2ψ1+2ψ2−4ψ3 → (1, 0, 0, -2, 2, -4)Vω1+2ψ1+2ψ2−4ψ3 → (1, 0, 0, 2, 2, -4)Vω1−2ψ1−2ψ2+4ψ3 → (1, 0, 0, -2, -2, 4)Vω1+2ψ1−2ψ2+4ψ3 → (1, 0, 0, 2, -2, 4)Vω2−2ψ1−2ψ2−2ψ3 → (0, 1, 0, -2, -2, -2)Vω2+2ψ1−2ψ2−2ψ3 → (0, 1, 0, 2, -2, -2)Vω2−2ψ1+2ψ2+2ψ3 → (0, 1, 0, -2, 2, 2)Vω2+2ψ1+2ψ2+2ψ3 → (0, 1, 0, 2, 2, 2)Vω3−2ψ1−4ψ2+2ψ3 → (0, 0, 1, -2, -4, 2)Vω3−2ψ1+4ψ2−2ψ3 → (0, 0, 1, -2, 4, -2)Vω3+2ψ1−4ψ2+2ψ3 → (0, 0, 1, 2, -4, 2)Vω3+2ψ1+4ψ2−2ψ3 → (0, 0, 1, 2, 4, -2)V2ω1 → (2, 0, 0, 0, 0, 0)Vω1+ω2−4ψ2+2ψ3 → (1, 1, 0, 0, -4, 2)Vω1+ω2+4ψ2−2ψ3 → (1, 1, 0, 0, 4, -2)V2ω2 → (0, 2, 0, 0, 0, 0)Vω1+ω3−2ψ2−2ψ3 → (1, 0, 1, 0, -2, -2)Vω1+ω3+2ψ2+2ψ3 → (1, 0, 1, 0, 2, 2)Vω2+ω3+2ψ2−4ψ3 → (0, 1, 1, 0, 2, -4)Vω2+ω3−2ψ2+4ψ3 → (0, 1, 1, 0, -2, 4)V2ω3 → (0, 0, 2, 0, 0, 0)Vω1+ω2+ω3−2ψ1 → (1, 1, 1, -2, 0, 0)Vω1+ω2+ω3+2ψ1 → (1, 1, 1, 2, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13W14W15W16W17W18W19W20W21W22W23W24W25W26
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g−4
Cartan of centralizer component.
−h5+h3
−h6+h1
h4
g4
g25
g−26
g28
−g−22
g22
g−28
g26
−g−25
g11
−g−12
g16
−g−7
g7
g−16
g12
g−11
g5
−g−9
g3
g−10
g10
−g−3
g9
g−5
Semisimple subalgebra component.
−g36
h6+2h5+3h4+2h3+2h2+h1
2g−36
g33
g−13
−g14
−g−32
g32
−g−14
g13
−g−33
Semisimple subalgebra component.
−g24
h6+h5+h4+h3+h1
2g−24
g31
−g−17
g20
−g−29
g29
g−20
−g17
−g−31
g21
−g−1
g6
−g−18
g18
g−6
−g1
−g−21
Semisimple subalgebra component.
−g15
h5+h4+h3
2g−15
g34
−g−8
−g19
−g27
g−30
g−23
g2
g−35
g35
−g−2
−g23
−g30
−g−27
−g−19
−g8
−g−34
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above000ω1
−ω1		ω1
−ω1		ω1
−ω1		ω1
−ω1		ω2
−ω2		ω2
−ω2		ω2
−ω2		ω2
−ω2		ω3
−ω3		ω3
−ω3		ω3
−ω3		ω3
−ω3		2ω1
0
−2ω1		ω1+ω2
−ω1+ω2
ω1−ω2
−ω1−ω2		ω1+ω2
−ω1+ω2
ω1−ω2
−ω1−ω2		2ω2
0
−2ω2		ω1+ω3
−ω1+ω3
ω1−ω3
−ω1−ω3		ω1+ω3
−ω1+ω3
ω1−ω3
−ω1−ω3		ω2+ω3
−ω2+ω3
ω2−ω3
−ω2−ω3		ω2+ω3
−ω2+ω3
ω2−ω3
−ω2−ω3		2ω3
0
−2ω3		ω1+ω2+ω3
−ω1+ω2+ω3
ω1−ω2+ω3
ω1+ω2−ω3
−ω1−ω2+ω3
−ω1+ω2−ω3
ω1−ω2−ω3
−ω1−ω2−ω3		ω1+ω2+ω3
−ω1+ω2+ω3
ω1−ω2+ω3
ω1+ω2−ω3
−ω1−ω2+ω3
−ω1+ω2−ω3
ω1−ω2−ω3
−ω1−ω2−ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer−4ψ104ψ1ω1−2ψ1+2ψ2−4ψ3
−ω1−2ψ1+2ψ2−4ψ3		ω1+2ψ1+2ψ2−4ψ3
−ω1+2ψ1+2ψ2−4ψ3		ω1−2ψ1−2ψ2+4ψ3
−ω1−2ψ1−2ψ2+4ψ3		ω1+2ψ1−2ψ2+4ψ3
−ω1+2ψ1−2ψ2+4ψ3		ω2−2ψ1−2ψ2−2ψ3
−ω2−2ψ1−2ψ2−2ψ3		ω2+2ψ1−2ψ2−2ψ3
−ω2+2ψ1−2ψ2−2ψ3		ω2−2ψ1+2ψ2+2ψ3
−ω2−2ψ1+2ψ2+2ψ3		ω2+2ψ1+2ψ2+2ψ3
−ω2+2ψ1+2ψ2+2ψ3		ω3−2ψ1−4ψ2+2ψ3
−ω3−2ψ1−4ψ2+2ψ3		ω3−2ψ1+4ψ2−2ψ3
−ω3−2ψ1+4ψ2−2ψ3		ω3+2ψ1−4ψ2+2ψ3
−ω3+2ψ1−4ψ2+2ψ3		ω3+2ψ1+4ψ2−2ψ3
−ω3+2ψ1+4ψ2−2ψ3		2ω1
0
−2ω1		ω1+ω2−4ψ2+2ψ3
−ω1+ω2−4ψ2+2ψ3
ω1−ω2−4ψ2+2ψ3
−ω1−ω2−4ψ2+2ψ3		ω1+ω2+4ψ2−2ψ3
−ω1+ω2+4ψ2−2ψ3
ω1−ω2+4ψ2−2ψ3
−ω1−ω2+4ψ2−2ψ3		2ω2
0
−2ω2		ω1+ω3−2ψ2−2ψ3
−ω1+ω3−2ψ2−2ψ3
ω1−ω3−2ψ2−2ψ3
−ω1−ω3−2ψ2−2ψ3		ω1+ω3+2ψ2+2ψ3
−ω1+ω3+2ψ2+2ψ3
ω1−ω3+2ψ2+2ψ3
−ω1−ω3+2ψ2+2ψ3		ω2+ω3+2ψ2−4ψ3
−ω2+ω3+2ψ2−4ψ3
ω2−ω3+2ψ2−4ψ3
−ω2−ω3+2ψ2−4ψ3		ω2+ω3−2ψ2+4ψ3
−ω2+ω3−2ψ2+4ψ3
ω2−ω3−2ψ2+4ψ3
−ω2−ω3−2ψ2+4ψ3		2ω3
0
−2ω3		ω1+ω2+ω3−2ψ1
−ω1+ω2+ω3−2ψ1
ω1−ω2+ω3−2ψ1
ω1+ω2−ω3−2ψ1
−ω1−ω2+ω3−2ψ1
−ω1+ω2−ω3−2ψ1
ω1−ω2−ω3−2ψ1
−ω1−ω2−ω3−2ψ1		ω1+ω2+ω3+2ψ1
−ω1+ω2+ω3+2ψ1
ω1−ω2+ω3+2ψ1
ω1+ω2−ω3+2ψ1
−ω1−ω2+ω3+2ψ1
−ω1+ω2−ω3+2ψ1
ω1−ω2−ω3+2ψ1
−ω1−ω2−ω3+2ψ1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M−4ψ1M0M4ψ1Mω1−2ψ1+2ψ2−4ψ3⊕M−ω1−2ψ1+2ψ2−4ψ3Mω1+2ψ1+2ψ2−4ψ3⊕M−ω1+2ψ1+2ψ2−4ψ3Mω1−2ψ1−2ψ2+4ψ3⊕M−ω1−2ψ1−2ψ2+4ψ3Mω1+2ψ1−2ψ2+4ψ3⊕M−ω1+2ψ1−2ψ2+4ψ3Mω2−2ψ1−2ψ2−2ψ3⊕M−ω2−2ψ1−2ψ2−2ψ3Mω2+2ψ1−2ψ2−2ψ3⊕M−ω2+2ψ1−2ψ2−2ψ3Mω2−2ψ1+2ψ2+2ψ3⊕M−ω2−2ψ1+2ψ2+2ψ3Mω2+2ψ1+2ψ2+2ψ3⊕M−ω2+2ψ1+2ψ2+2ψ3Mω3−2ψ1−4ψ2+2ψ3⊕M−ω3−2ψ1−4ψ2+2ψ3Mω3−2ψ1+4ψ2−2ψ3⊕M−ω3−2ψ1+4ψ2−2ψ3Mω3+2ψ1−4ψ2+2ψ3⊕M−ω3+2ψ1−4ψ2+2ψ3Mω3+2ψ1+4ψ2−2ψ3⊕M−ω3+2ψ1+4ψ2−2ψ3M2ω1⊕M0⊕M−2ω1Mω1+ω2−4ψ2+2ψ3⊕M−ω1+ω2−4ψ2+2ψ3⊕Mω1−ω2−4ψ2+2ψ3⊕M−ω1−ω2−4ψ2+2ψ3Mω1+ω2+4ψ2−2ψ3⊕M−ω1+ω2+4ψ2−2ψ3⊕Mω1−ω2+4ψ2−2ψ3⊕M−ω1−ω2+4ψ2−2ψ3M2ω2⊕M0⊕M−2ω2Mω1+ω3−2ψ2−2ψ3⊕M−ω1+ω3−2ψ2−2ψ3⊕Mω1−ω3−2ψ2−2ψ3⊕M−ω1−ω3−2ψ2−2ψ3Mω1+ω3+2ψ2+2ψ3⊕M−ω1+ω3+2ψ2+2ψ3⊕Mω1−ω3+2ψ2+2ψ3⊕M−ω1−ω3+2ψ2+2ψ3Mω2+ω3+2ψ2−4ψ3⊕M−ω2+ω3+2ψ2−4ψ3⊕Mω2−ω3+2ψ2−4ψ3⊕M−ω2−ω3+2ψ2−4ψ3Mω2+ω3−2ψ2+4ψ3⊕M−ω2+ω3−2ψ2+4ψ3⊕Mω2−ω3−2ψ2+4ψ3⊕M−ω2−ω3−2ψ2+4ψ3M2ω3⊕M0⊕M−2ω3Mω1+ω2+ω3−2ψ1⊕M−ω1+ω2+ω3−2ψ1⊕Mω1−ω2+ω3−2ψ1⊕Mω1+ω2−ω3−2ψ1⊕M−ω1−ω2+ω3−2ψ1⊕M−ω1+ω2−ω3−2ψ1⊕Mω1−ω2−ω3−2ψ1⊕M−ω1−ω2−ω3−2ψ1Mω1+ω2+ω3+2ψ1⊕M−ω1+ω2+ω3+2ψ1⊕Mω1−ω2+ω3+2ψ1⊕Mω1+ω2−ω3+2ψ1⊕M−ω1−ω2+ω3+2ψ1⊕M−ω1+ω2−ω3+2ψ1⊕Mω1−ω2−ω3+2ψ1⊕M−ω1−ω2−ω3+2ψ1
Isotypic characterM−4ψ13M0M4ψ1Mω1−2ψ1+2ψ2−4ψ3⊕M−ω1−2ψ1+2ψ2−4ψ3Mω1+2ψ1+2ψ2−4ψ3⊕M−ω1+2ψ1+2ψ2−4ψ3Mω1−2ψ1−2ψ2+4ψ3⊕M−ω1−2ψ1−2ψ2+4ψ3Mω1+2ψ1−2ψ2+4ψ3⊕M−ω1+2ψ1−2ψ2+4ψ3Mω2−2ψ1−2ψ2−2ψ3⊕M−ω2−2ψ1−2ψ2−2ψ3Mω2+2ψ1−2ψ2−2ψ3⊕M−ω2+2ψ1−2ψ2−2ψ3Mω2−2ψ1+2ψ2+2ψ3⊕M−ω2−2ψ1+2ψ2+2ψ3Mω2+2ψ1+2ψ2+2ψ3⊕M−ω2+2ψ1+2ψ2+2ψ3Mω3−2ψ1−4ψ2+2ψ3⊕M−ω3−2ψ1−4ψ2+2ψ3Mω3−2ψ1+4ψ2−2ψ3⊕M−ω3−2ψ1+4ψ2−2ψ3Mω3+2ψ1−4ψ2+2ψ3⊕M−ω3+2ψ1−4ψ2+2ψ3Mω3+2ψ1+4ψ2−2ψ3⊕M−ω3+2ψ1+4ψ2−2ψ3M2ω1⊕M0⊕M−2ω1Mω1+ω2−4ψ2+2ψ3⊕M−ω1+ω2−4ψ2+2ψ3⊕Mω1−ω2−4ψ2+2ψ3⊕M−ω1−ω2−4ψ2+2ψ3Mω1+ω2+4ψ2−2ψ3⊕M−ω1+ω2+4ψ2−2ψ3⊕Mω1−ω2+4ψ2−2ψ3⊕M−ω1−ω2+4ψ2−2ψ3M2ω2⊕M0⊕M−2ω2Mω1+ω3−2ψ2−2ψ3⊕M−ω1+ω3−2ψ2−2ψ3⊕Mω1−ω3−2ψ2−2ψ3⊕M−ω1−ω3−2ψ2−2ψ3Mω1+ω3+2ψ2+2ψ3⊕M−ω1+ω3+2ψ2+2ψ3⊕Mω1−ω3+2ψ2+2ψ3⊕M−ω1−ω3+2ψ2+2ψ3Mω2+ω3+2ψ2−4ψ3⊕M−ω2+ω3+2ψ2−4ψ3⊕Mω2−ω3+2ψ2−4ψ3⊕M−ω2−ω3+2ψ2−4ψ3Mω2+ω3−2ψ2+4ψ3⊕M−ω2+ω3−2ψ2+4ψ3⊕Mω2−ω3−2ψ2+4ψ3⊕M−ω2−ω3−2ψ2+4ψ3M2ω3⊕M0⊕M−2ω3Mω1+ω2+ω3−2ψ1⊕M−ω1+ω2+ω3−2ψ1⊕Mω1−ω2+ω3−2ψ1⊕Mω1+ω2−ω3−2ψ1⊕M−ω1−ω2+ω3−2ψ1⊕M−ω1+ω2−ω3−2ψ1⊕Mω1−ω2−ω3−2ψ1⊕M−ω1−ω2−ω3−2ψ1Mω1+ω2+ω3+2ψ1⊕M−ω1+ω2+ω3+2ψ1⊕Mω1−ω2+ω3+2ψ1⊕Mω1+ω2−ω3+2ψ1⊕M−ω1−ω2+ω3+2ψ1⊕M−ω1+ω2−ω3+2ψ1⊕Mω1−ω2−ω3+2ψ1⊕M−ω1−ω2−ω3+2ψ1

Semisimple subalgebra: W_{16}+W_{19}+W_{24}
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.
Canvas not supported




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)
(0.00, 1.00, 0.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00, 0.00): (250.00, 300.00)
1: (0.00, 1.00, 0.00, 0.00, 0.00, 0.00): (200.00, 350.00)
2: (0.00, 0.00, 1.00, 0.00, 0.00, 0.00): (200.00, 300.00)
3: (0.00, 0.00, 0.00, 1.00, 0.00, 0.00): (200.00, 300.00)
4: (0.00, 0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
5: (0.00, 0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 300.00)



Made total 460 arithmetic operations while solving the Serre relations polynomial system.