Subalgebra \(A^{35}_1\) ↪ \(A^{1}_5\)
10 out of 37

Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{35}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (5, 8, 9, 8, 5): 70
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle 5g_{5}+8g_{4}+9g_{3}+8g_{2}+5g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\)
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 5) ; the vectors are over the primal subalgebra.	\(g_{5}+8/5g_{4}+9/5g_{3}+8/5g_{2}+g_{1}\)	\(g_{9}+9/5g_{8}+9/5g_{7}+g_{6}\)	\(g_{12}+8/5g_{11}+g_{10}\)	\(g_{14}+g_{13}\)	\(g_{15}\)
weight	\(2\omega_{1}\)	\(4\omega_{1}\)	\(6\omega_{1}\)	\(8\omega_{1}\)	\(10\omega_{1}\)

Isotypic module decomposition over primal subalgebra (total 5 isotypic components).

Isotypical components + highest weight

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

\(\displaystyle V_{4\omega_{1}} \) → (4)

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

\(\displaystyle V_{8\omega_{1}} \) → (8)

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

Module label

\(W_{1}\)

\(W_{2}\)

\(W_{3}\)

\(W_{4}\)

\(W_{5}\)

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

Semisimple subalgebra component.

\(-g_{5}-8/5g_{4}-9/5g_{3}-8/5g_{2}-g_{1}\)

\(h_{5}+8/5h_{4}+9/5h_{3}+8/5h_{2}+h_{1}\)

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

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

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

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

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

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

\(g_{12}+8/5g_{11}+g_{10}\)

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

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

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

\(-12/5g_{-1}+3/5g_{-2}+8/5g_{-3}+3/5g_{-4}-12/5g_{-5}\)

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

\(-2g_{-10}-2g_{-11}-2g_{-12}\)

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

\(g_{12}-g_{10}\)

\(g_{9}-g_{8}-g_{7}+g_{6}\)

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

\(-h_{5}+2h_{4}-2h_{2}+h_{1}\)

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

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

\(14g_{-10}-14g_{-12}\)

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

\(g_{15}\)

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

\(g_{12}-2g_{11}+g_{10}\)

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

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

\(-h_{5}+4h_{4}-6h_{3}+4h_{2}-h_{1}\)

\(-6g_{-1}+15g_{-2}-20g_{-3}+15g_{-4}-6g_{-5}\)

\(-21g_{-6}+35g_{-7}-35g_{-8}+21g_{-9}\)

\(-56g_{-10}+70g_{-11}-56g_{-12}\)

\(-126g_{-13}+126g_{-14}\)

\(-252g_{-15}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Isotypic character

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

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

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

Semisimple subalgebra: W_{1}
Centralizer extension: 0

Made total 56408 arithmetic operations while solving the Serre relations polynomial system.