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Subalgebra

$A^{10}_1+A^{2}_1$ ↪

$D^{1}_4$
12 out of 23

Computations done by the calculator project.

Subalgebra type:

$\displaystyle A^{10}_1+A^{2}_1$ (click on type for detailed printout).
Subalgebra is (parabolically) induced from

$\displaystyle A^{10}_1$ .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer:

$\displaystyle D^{1}_4$

Elements Cartan subalgebra scaled to act by two by components:

$\displaystyle A^{10}_1$ : (3, 6, 3, 4): 20,

$\displaystyle A^{2}_1$ : (1, 0, 1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators:

$\displaystyle g_{-4}+g_{-5}+g_{-6}$ ,

$\displaystyle g_{-1}+g_{-3}$
Positive simple generators:

$\displaystyle 3g_{6}+3g_{5}+4g_{4}$ ,

$\displaystyle g_{3}+g_{1}$
Cartan symmetric matrix:

$\displaystyle \begin{pmatrix}1/5 & 0\\ 0 & 1\\ \end{pmatrix}$
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix):

$\displaystyle \begin{pmatrix}20 & 0\\ 0 & 4\\ \end{pmatrix}$
Decomposition of ambient Lie algebra:

$\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}$
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 4) ; the vectors are over the primal subalgebra.	$g_{6}+g_{5}+4/3g_{4}$	$g_{3}+g_{1}$	$g_{12}$	$g_{11}$
weight	$2\omega_{1}$	$2\omega_{2}$	$6\omega_{1}$	$4\omega_{1}+2\omega_{2}$

Isotypic module decomposition over primal subalgebra (total 4 isotypic components).

Isotypical components + highest weight

$\displaystyle V_{2\omega_{1}}$ → (2, 0)

$\displaystyle V_{2\omega_{2}}$ → (0, 2)

$\displaystyle V_{6\omega_{1}}$ → (6, 0)

$\displaystyle V_{4\omega_{1}+2\omega_{2}}$ → (4, 2)

Module label

$W_{1}$

$W_{2}$

$W_{3}$

$W_{4}$

Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.

Semisimple subalgebra component.

$-3/4g_{6}-3/4g_{5}-g_{4}$

$h_{4}+3/4h_{3}+3/2h_{2}+3/4h_{1}$

$1/2g_{-4}+1/2g_{-5}+1/2g_{-6}$

Semisimple subalgebra component.

$-g_{3}-g_{1}$

$h_{3}+h_{1}$

$2g_{-1}+2g_{-3}$

$g_{12}$

$-g_{10}+g_{9}$

$-g_{6}-g_{5}+2g_{4}$

$-2h_{4}+h_{3}+2h_{2}+h_{1}$

$-6g_{-4}+4g_{-5}+4g_{-6}$

$10g_{-9}-10g_{-10}$

$-20g_{-12}$

$g_{11}$

$g_{8}$

$g_{10}+g_{9}$

$g_{3}-g_{1}$

$g_{6}-g_{5}$

$2g_{7}$

$-2g_{-2}$

$-h_{3}+h_{1}$

$-2g_{2}$

$-2g_{-7}$

$2g_{-5}-2g_{-6}$

$2g_{-1}-2g_{-3}$

$2g_{-9}+2g_{-10}$

$4g_{-8}$

$-4g_{-11}$

Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above

$2\omega_{1}$

$0$

$-2\omega_{1}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$4\omega_{1}+2\omega_{2}$

$2\omega_{1}+2\omega_{2}$

$4\omega_{1}$

$2\omega_{2}$

$2\omega_{1}$

$4\omega_{1}-2\omega_{2}$

$-2\omega_{1}+2\omega_{2}$

$0$

$2\omega_{1}-2\omega_{2}$

$-4\omega_{1}+2\omega_{2}$

$-2\omega_{1}$

$-2\omega_{2}$

$-4\omega_{1}$

$-2\omega_{1}-2\omega_{2}$

$-4\omega_{1}-2\omega_{2}$

Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer

$2\omega_{1}$

$0$

$-2\omega_{1}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$4\omega_{1}+2\omega_{2}$

$2\omega_{1}+2\omega_{2}$

$4\omega_{1}$

$2\omega_{2}$

$2\omega_{1}$

$4\omega_{1}-2\omega_{2}$

$-2\omega_{1}+2\omega_{2}$

$0$

$2\omega_{1}-2\omega_{2}$

$-4\omega_{1}+2\omega_{2}$

$-2\omega_{1}$

$-2\omega_{2}$

$-4\omega_{1}$

$-2\omega_{1}-2\omega_{2}$

$-4\omega_{1}-2\omega_{2}$

Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.

$\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}$

$\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}$

$\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}$

$\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}} \oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}} \oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}$

Isotypic character

$\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}$

$\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}$

$\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}$

Semisimple subalgebra: W_{1}+W_{2}
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.

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, 1.00)
0: (1.00, 0.00): (700.00, 300.00)
1: (0.00, 1.00): (200.00, 400.00)

Made total 683185 arithmetic operations while solving the Serre relations polynomial system.