Processing math: 100%

Subalgebra

$A^{156}_1$ ↪

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

Computations done by the calculator project.

Subalgebra type:

$\displaystyle A^{156}_1$ (click on type for detailed printout).
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:

$\displaystyle A^{156}_1$ : (22, 42, 60, 32): 312
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators:

$\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}$
Positive simple generators:

$\displaystyle 16g_{4}+30g_{3}+42g_{2}+22g_{1}$
Cartan symmetric matrix:

$\displaystyle \begin{pmatrix}1/78\\ \end{pmatrix}$
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix):

$\displaystyle \begin{pmatrix}312\\ \end{pmatrix}$
Decomposition of ambient Lie algebra:

$\displaystyle V_{22\omega_{1}}\oplus V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{2\omega_{1}}$
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.	$g_{4}+15/8g_{3}+21/8g_{2}+11/8g_{1}$	$-g_{16}+11/16g_{15}+231/128g_{14}$	$-g_{20}+15/8g_{19}$	$g_{24}$
weight	$2\omega_{1}$	$10\omega_{1}$	$14\omega_{1}$	$22\omega_{1}$

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

Isotypical components + highest weight

$\displaystyle V_{2\omega_{1}}$ → (2)

$\displaystyle V_{10\omega_{1}}$ → (10)

$\displaystyle V_{14\omega_{1}}$ → (14)

$\displaystyle V_{22\omega_{1}}$ → (22)

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.

$-8/11g_{4}-15/11g_{3}-21/11g_{2}-g_{1}$

$16/11h_{4}+30/11h_{3}+21/11h_{2}+h_{1}$

$1/11g_{-1}+1/11g_{-2}+1/11g_{-3}+1/11g_{-4}$

$-g_{16}+11/16g_{15}+231/128g_{14}$

$-5/16g_{13}+11/16g_{12}+55/128g_{11}$

$3/8g_{10}+135/128g_{9}-33/128g_{8}$

$3/8g_{7}+27/64g_{6}+33/64g_{5}$

$3/8g_{4}+3/64g_{3}-21/64g_{2}-33/64g_{1}$

$-3/4h_{4}-3/32h_{3}+21/64h_{2}+33/64h_{1}$

$45/64g_{-1}+15/64g_{-2}-3/64g_{-3}-45/64g_{-4}$

$15/32g_{-5}+9/32g_{-6}+21/32g_{-7}$

$3/16g_{-8}-9/16g_{-9}-3/8g_{-10}$

$3/16g_{-11}+9/16g_{-12}-3/16g_{-13}$

$-3/16g_{-14}-3/16g_{-15}+3/8g_{-16}$

$-g_{20}+15/8g_{19}$

$-g_{18}+7/8g_{17}$

$-g_{16}-1/8g_{15}-7/4g_{14}$

$-9/8g_{13}-1/8g_{12}-3/2g_{11}$

$-5/4g_{10}+3/4g_{9}-11/8g_{8}$

$-5/4g_{7}+5/8g_{6}+11/4g_{5}$

$-5/4g_{4}+15/8g_{3}+3/2g_{2}-11/4g_{1}$

$5/2h_{4}-15/4h_{3}-3/2h_{2}+11/4h_{1}$

$7g_{-1}-2g_{-2}-7/2g_{-3}+35/8g_{-4}$

$9g_{-5}+3/2g_{-6}-63/8g_{-7}$

$15/2g_{-8}-3g_{-9}+75/8g_{-10}$

$-12g_{-11}-15/8g_{-12}-99/8g_{-13}$

$12g_{-14}+9/4g_{-15}+99/4g_{-16}$

$39/4g_{-17}-117/4g_{-18}$

$-39/4g_{-19}+39/4g_{-20}$

$g_{24}$

$g_{23}$

$g_{22}$

$g_{21}$

$2g_{20}+g_{19}$

$2g_{18}+3g_{17}$

$2g_{16}+5g_{15}-6g_{14}$

$7g_{13}+5g_{12}-16g_{11}$

$12g_{10}-30g_{9}-21g_{8}$

$12g_{7}-63g_{6}+42g_{5}$

$12g_{4}-75g_{3}+168g_{2}-42g_{1}$

$-24h_{4}+150h_{3}-168h_{2}+42h_{1}$

$252g_{-1}-528g_{-2}+330g_{-3}-99g_{-4}$

$780g_{-5}-858g_{-6}+429g_{-7}$

$1638g_{-8}+1716g_{-9}-1287g_{-10}$

$-4992g_{-11}+2925g_{-12}+3003g_{-13}$

$4992g_{-14}-10920g_{-15}-6006g_{-16}$

$15912g_{-17}+27846g_{-18}$

$-15912g_{-19}-59670g_{-20}$

$75582g_{-21}$

$-151164g_{-22}$

$151164g_{-23}$

$-151164g_{-24}$

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

$10\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

$-10\omega_{1}$

$14\omega_{1}$

$12\omega_{1}$

$10\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

$-10\omega_{1}$

$-12\omega_{1}$

$-14\omega_{1}$

$22\omega_{1}$

$20\omega_{1}$

$18\omega_{1}$

$16\omega_{1}$

$14\omega_{1}$

$12\omega_{1}$

$10\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

$-10\omega_{1}$

$-12\omega_{1}$

$-14\omega_{1}$

$-16\omega_{1}$

$-18\omega_{1}$

$-20\omega_{1}$

$-22\omega_{1}$

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

$10\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

$-10\omega_{1}$

$14\omega_{1}$

$12\omega_{1}$

$10\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

$-10\omega_{1}$

$-12\omega_{1}$

$-14\omega_{1}$

$22\omega_{1}$

$20\omega_{1}$

$18\omega_{1}$

$16\omega_{1}$

$14\omega_{1}$

$12\omega_{1}$

$10\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

$-10\omega_{1}$

$-12\omega_{1}$

$-14\omega_{1}$

$-16\omega_{1}$

$-18\omega_{1}$

$-20\omega_{1}$

$-22\omega_{1}$

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

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

$\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}} \oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}$

$\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}} \oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}} \oplus M_{-14\omega_{1}}$

$\displaystyle M_{22\omega_{1}}\oplus M_{20\omega_{1}}\oplus M_{18\omega_{1}}\oplus M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}} \oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}} \oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\oplus M_{-18\omega_{1}} \oplus M_{-20\omega_{1}}\oplus M_{-22\omega_{1}}$

Isotypic character

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

Semisimple subalgebra: W_{1}
Centralizer extension: 0

Made total 21930 arithmetic operations while solving the Serre relations polynomial system.