Subalgebra \(A^{84}_1\) ↪ \(C^{1}_4\)
13 out of 46

Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{84}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{84}_1\): (14, 24, 30, 16): 168
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: \(\displaystyle 16g_{4}+15g_{3}+12g_{2}+7g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/42\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}168\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{1}}\)
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.	\(g_{4}+15/16g_{3}+3/4g_{2}+7/16g_{1}\)	\(-g_{10}+4/5g_{9}+7/20g_{8}\)	\(-g_{14}+7/12g_{13}\)	\(g_{16}\)
weight	\(2\omega_{1}\)	\(6\omega_{1}\)	\(10\omega_{1}\)	\(14\omega_{1}\)

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

Isotypical components + highest weight

\(\displaystyle V_{2\omega_{1}} \) → (2)

\(\displaystyle V_{6\omega_{1}} \) → (6)

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

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

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.

\(-16/7g_{4}-15/7g_{3}-12/7g_{2}-g_{1}\)

\(16/7h_{4}+30/7h_{3}+24/7h_{2}+2h_{1}\)

\(2/7g_{-1}+2/7g_{-2}+2/7g_{-3}+2/7g_{-4}\)

\(-g_{10}+4/5g_{9}+7/20g_{8}\)

\(-1/5g_{7}-9/20g_{6}-7/20g_{5}\)

\(-2/5g_{4}-1/4g_{3}+1/10g_{2}+7/20g_{1}\)

\(2/5h_{4}+1/2h_{3}-1/5h_{2}-7/10h_{1}\)

\(-3/5g_{-1}-1/10g_{-2}+1/5g_{-3}+3/10g_{-4}\)

\(-1/2g_{-5}-3/10g_{-6}-1/10g_{-7}\)

\(-1/5g_{-8}-1/5g_{-9}+1/5g_{-10}\)

\(-g_{14}+7/12g_{13}\)

\(-5/12g_{12}+7/12g_{11}\)

\(-5/6g_{10}+1/6g_{9}-7/12g_{8}\)

\(-2/3g_{7}-3/4g_{6}+7/12g_{5}\)

\(-4/3g_{4}-1/12g_{3}+4/3g_{2}-7/12g_{1}\)

\(4/3h_{4}+1/6h_{3}-8/3h_{2}+7/6h_{1}\)

\(5/2g_{-1}-10/3g_{-2}+1/6g_{-3}+5/2g_{-4}\)

\(35/6g_{-5}-7/2g_{-6}-7/3g_{-7}\)

\(28/3g_{-8}-7/6g_{-9}+14/3g_{-10}\)

\(21/2g_{-11}-7/2g_{-12}\)

\(-7g_{-13}+7g_{-14}\)

\(g_{16}\)

\(g_{15}\)

\(2g_{14}+g_{13}\)

\(3g_{12}+g_{11}\)

\(6g_{10}+4g_{9}-g_{8}\)

\(10g_{7}-5g_{6}+g_{5}\)

\(20g_{4}-15g_{3}+6g_{2}-g_{1}\)

\(-20h_{4}+30h_{3}-12h_{2}+2h_{1}\)

\(8g_{-1}-28g_{-2}+56g_{-3}-70g_{-4}\)

\(36g_{-5}-84g_{-6}+126g_{-7}\)

\(120g_{-8}-210g_{-9}-252g_{-10}\)

\(330g_{-11}+462g_{-12}\)

\(-792g_{-13}-924g_{-14}\)

\(1716g_{-15}\)

\(-3432g_{-16}\)

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

\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)

\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)

\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)

\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)

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

\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)

\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)

\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)

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

\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)

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

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

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

Isotypic character

\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)

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

Semisimple subalgebra: W_{1}
Centralizer extension: 0

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