Subalgebra \(A^{60}_1\) ↪ \(B^{1}_4\)
12 out of 48

Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{60}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{60}_1\): (8, 14, 18, 20): 120
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: \(\displaystyle 10g_{4}+18g_{3}+14g_{2}+8g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/30\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}120\\ \end{pmatrix}\)
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.	\(g_{4}+9/5g_{3}+7/5g_{2}+4/5g_{1}\)	\(-g_{10}+7/10g_{9}+14/25g_{8}\)	\(-g_{14}+4/9g_{13}\)	\(g_{16}\)
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.

\(-5/4g_{4}-9/4g_{3}-7/4g_{2}-g_{1}\)

\(5/2h_{4}+9/4h_{3}+7/4h_{2}+h_{1}\)

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

\(-g_{10}+7/10g_{9}+14/25g_{8}\)

\(-3/10g_{7}-21/25g_{6}-14/25g_{5}\)

\(-3/10g_{4}-6/25g_{3}+7/25g_{2}+14/25g_{1}\)

\(3/5h_{4}+6/25h_{3}-7/25h_{2}-14/25h_{1}\)

\(-21/25g_{-1}-6/25g_{-2}+4/25g_{-3}+9/25g_{-4}\)

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

\(-1/5g_{-8}-1/5g_{-9}+2/5g_{-10}\)

\(-g_{14}+4/9g_{13}\)

\(-5/9g_{12}+4/9g_{11}\)

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

\(-2/3g_{7}-2/3g_{6}+8/9g_{5}\)

\(-2/3g_{4}+2/3g_{3}+14/9g_{2}-8/9g_{1}\)

\(4/3h_{4}-2/3h_{3}-14/9h_{2}+8/9h_{1}\)

\(10/3g_{-1}-10/3g_{-2}-10/9g_{-3}+2g_{-4}\)

\(20/3g_{-5}-20/9g_{-6}-28/9g_{-7}\)

\(80/9g_{-8}+8/9g_{-9}+56/9g_{-10}\)

\(8g_{-11}-8g_{-12}\)

\(-8g_{-13}+8g_{-14}\)

\(g_{16}\)

\(g_{15}\)

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

\(2g_{12}+g_{11}\)

\(2g_{10}+3g_{9}-2g_{8}\)

\(5g_{7}-8g_{6}+2g_{5}\)

\(5g_{4}-18g_{3}+10g_{2}-2g_{1}\)

\(-10h_{4}+18h_{3}-10h_{2}+2h_{1}\)

\(14g_{-1}-40g_{-2}+56g_{-3}-28g_{-4}\)

\(54g_{-5}-96g_{-6}+84g_{-7}\)

\(150g_{-8}-180g_{-9}-168g_{-10}\)

\(330g_{-11}+528g_{-12}\)

\(-1188g_{-13}-528g_{-14}\)

\(1716g_{-15}\)

\(-1716g_{-16}\)

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.