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Subalgebra

$3A^{1}_1$ ↪

$C^{1}_3$
14 out of 16

Computations done by the calculator project.

Subalgebra type:

$\displaystyle 3A^{1}_1$ (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from

$\displaystyle 2A^{1}_1$ .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer:

$\displaystyle C^{1}_3$

Elements Cartan subalgebra scaled to act by two by components:

$\displaystyle A^{1}_1$ : (2, 2, 1): 2,

$\displaystyle A^{1}_1$ : (0, 2, 1): 2,

$\displaystyle A^{1}_1$ : (0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators:

$\displaystyle g_{-9}$ ,

$\displaystyle g_{-7}$ ,

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

$\displaystyle g_{9}$ ,

$\displaystyle g_{7}$ ,

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

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

$\displaystyle \begin{pmatrix}2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Decomposition of ambient Lie algebra:

$\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{2\omega_{1}}$
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 6) ; the vectors are over the primal subalgebra.	$g_{9}$	$g_{8}$	$g_{7}$	$g_{6}$	$g_{5}$	$g_{3}$
weight	$2\omega_{1}$	$\omega_{1}+\omega_{2}$	$2\omega_{2}$	$\omega_{1}+\omega_{3}$	$\omega_{2}+\omega_{3}$	$2\omega_{3}$

Isotypic module decomposition over primal subalgebra (total 6 isotypic components).

Isotypical components + highest weight

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

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

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

$\displaystyle V_{\omega_{1}+\omega_{3}}$ → (1, 0, 1)

$\displaystyle V_{\omega_{2}+\omega_{3}}$ → (0, 1, 1)

$\displaystyle V_{2\omega_{3}}$ → (0, 0, 2)

Module label

$W_{1}$

$W_{2}$

$W_{3}$

$W_{4}$

$W_{5}$

$W_{6}$

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

Semisimple subalgebra component.

$-g_{9}$

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

$2g_{-9}$

$g_{8}$

$-g_{-1}$

$-g_{1}$

$-g_{-8}$

Semisimple subalgebra component.

$-g_{7}$

$h_{3}+2h_{2}$

$2g_{-7}$

$g_{6}$

$-g_{-4}$

$-g_{4}$

$-g_{-6}$

$g_{5}$

$-g_{-2}$

$-g_{2}$

$-g_{-5}$

Semisimple subalgebra component.

$-g_{3}$

$h_{3}$

$2g_{-3}$

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

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

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

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

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

$2\omega_{2}$

$0$

$-2\omega_{2}$

$\omega_{1}+\omega_{3}$

$-\omega_{1}+\omega_{3}$

$\omega_{1}-\omega_{3}$

$-\omega_{1}-\omega_{3}$

$\omega_{2}+\omega_{3}$

$-\omega_{2}+\omega_{3}$

$\omega_{2}-\omega_{3}$

$-\omega_{2}-\omega_{3}$

$2\omega_{3}$

$0$

$-2\omega_{3}$

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

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

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

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

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

$2\omega_{2}$

$0$

$-2\omega_{2}$

$\omega_{1}+\omega_{3}$

$-\omega_{1}+\omega_{3}$

$\omega_{1}-\omega_{3}$

$-\omega_{1}-\omega_{3}$

$\omega_{2}+\omega_{3}$

$-\omega_{2}+\omega_{3}$

$\omega_{2}-\omega_{3}$

$-\omega_{2}-\omega_{3}$

$2\omega_{3}$

$0$

$-2\omega_{3}$

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

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

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

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

$\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}-\omega_{3}}$

$\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}$

$\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}$

Isotypic character

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

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

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

$\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}-\omega_{3}}$

$\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}$

$\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}$

Semisimple subalgebra: W_{1}+W_{3}+W_{6}
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)
(0.00, 1.00, 0.00)
0: (1.00, 0.00, 0.00): (250.00, 300.00)
1: (0.00, 1.00, 0.00): (200.00, 350.00)
2: (0.00, 0.00, 1.00): (200.00, 300.00)

Made total 444 arithmetic operations while solving the Serre relations polynomial system.