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Subalgebra

$A^{10}_1+A^{2}_1$ ↪

$F^{1}_4$
27 out of 59

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:

$\displaystyle F^{1}_4$

Elements Cartan subalgebra scaled to act by two by components:

$\displaystyle A^{10}_1$ : (6, 10, 14, 8): 20,

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

$\displaystyle g_{-5}+g_{-11}+g_{-16}$ ,

$\displaystyle -g_{-2}+g_{-9}$
Positive simple generators:

$\displaystyle 4g_{16}+3g_{11}+3g_{5}$ ,

$\displaystyle g_{9}-g_{2}$
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:

$\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{2}}\oplus V_{2\omega_{1}}$
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 8) ; the vectors are over the primal subalgebra.	$g_{16}+3/4g_{11}+3/4g_{5}$	$g_{6}$	$-g_{9}+g_{2}$	$g_{21}$	$g_{17}$	$g_{19}$	$g_{24}$	$g_{23}$
weight	$2\omega_{1}$	$2\omega_{2}$	$2\omega_{2}$	$4\omega_{1}$	$3\omega_{1}+\omega_{2}$	$3\omega_{1}+\omega_{2}$	$6\omega_{1}$	$4\omega_{1}+2\omega_{2}$

Isotypic module decomposition over primal subalgebra (total 7 isotypic components).

Isotypical components + highest weight

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

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

$\displaystyle V_{4\omega_{1}}$ → (4, 0)

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

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

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

Module label

$W_{1}$

$W_{2}$

$W_{3}$

$W_{4}$

$W_{5}$

$W_{6}$

$W_{7}$

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

Semisimple subalgebra component.

$-4/3g_{16}-g_{11}-g_{5}$

$8/3h_{4}+14/3h_{3}+10/3h_{2}+2h_{1}$

$2/3g_{-5}+2/3g_{-11}+2/3g_{-16}$

Semisimple subalgebra component.

$g_{9}-g_{2}$

$-2h_{3}-2h_{2}$

$2g_{-2}-2g_{-9}$

$g_{6}$

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

$-2g_{-6}$

$g_{21}$

$-g_{8}$

$-g_{3}+g_{-3}$

$-2g_{-8}$

$-2g_{-21}$

$g_{17}$

$-g_{13}$

$-g_{15}$

$g_{-4}$

$-g_{4}$

$-g_{-15}$

$-g_{-13}$

$-g_{-17}$

$g_{19}$

$-g_{10}$

$g_{12}$

$-g_{-7}$

$g_{7}$

$g_{-12}$

$-g_{-10}$

$-g_{-19}$

$g_{24}$

$g_{22}-g_{20}$

$2g_{16}-g_{11}-g_{5}$

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

$4g_{-5}+4g_{-11}-6g_{-16}$

$-10g_{-20}+10g_{-22}$

$-20g_{-24}$

$g_{23}$

$-g_{14}$

$-g_{22}-g_{20}$

$g_{9}+g_{2}$

$g_{11}-g_{5}$

$2g_{18}$

$-2g_{-1}$

$-2h_{3}$

$-2g_{1}$

$-2g_{-18}$

$2g_{-5}-2g_{-11}$

$-2g_{-2}-2g_{-9}$

$-2g_{-20}-2g_{-22}$

$-4g_{-14}$

$-4g_{-23}$

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$3\omega_{1}+\omega_{2}$

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

$3\omega_{1}-\omega_{2}$

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

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

$-3\omega_{1}+\omega_{2}$

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

$-3\omega_{1}-\omega_{2}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

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

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

$4\omega_{1}$

$2\omega_{2}$

$2\omega_{1}$

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

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

$0$

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

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

$-2\omega_{1}$

$-2\omega_{2}$

$-4\omega_{1}$

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

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

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$3\omega_{1}+\omega_{2}$

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

$3\omega_{1}-\omega_{2}$

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

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

$-3\omega_{1}+\omega_{2}$

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

$-3\omega_{1}-\omega_{2}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

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

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

$4\omega_{1}$

$2\omega_{2}$

$2\omega_{1}$

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

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

$0$

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

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

$-2\omega_{1}$

$-2\omega_{2}$

$-4\omega_{1}$

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

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

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

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

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

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

$\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}} \oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}$

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

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

Isotypic character

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

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

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

$\displaystyle 2M_{3\omega_{1}+\omega_{2}}\oplus 2M_{\omega_{1}+\omega_{2}}\oplus 2M_{3\omega_{1}-\omega_{2}}\oplus 2M_{-\omega_{1}+\omega_{2}} \oplus 2M_{\omega_{1}-\omega_{2}}\oplus 2M_{-3\omega_{1}+\omega_{2}}\oplus 2M_{-\omega_{1}-\omega_{2}}\oplus 2M_{-3\omega_{1}-\omega_{2}}$

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

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 58015 arithmetic operations while solving the Serre relations polynomial system.