Subalgebra \(A^{165}_1\) ↪ \(C^{1}_5\)
23 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{165}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{165}_1\): (18, 32, 42, 48, 25): 330
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: \(\displaystyle 25g_{5}+24g_{4}+21g_{3}+16g_{2}+9g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/165\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}330\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{18\omega_{1}}\oplus V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{1}}\)
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 5) ; the vectors are over the primal subalgebra.\(g_{5}+24/25g_{4}+21/25g_{3}+16/25g_{2}+9/25g_{1}\)\(-g_{13}+7/8g_{12}+14/25g_{11}+21/100g_{10}\)\(g_{19}-16/21g_{18}+2/7g_{17}\)\(-g_{23}+9/16g_{22}\)\(g_{25}\)
weight\(2\omega_{1}\)\(6\omega_{1}\)\(10\omega_{1}\)\(14\omega_{1}\)\(18\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 5 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)\(\displaystyle V_{18\omega_{1}} \) → (18)
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.
\(-25/9g_{5}-8/3g_{4}-7/3g_{3}-16/9g_{2}-g_{1}\)
\(25/9h_{5}+16/3h_{4}+14/3h_{3}+32/9h_{2}+2h_{1}\)
\(2/9g_{-1}+2/9g_{-2}+2/9g_{-3}+2/9g_{-4}+2/9g_{-5}\)
\(-g_{13}+7/8g_{12}+14/25g_{11}+21/100g_{10}\)
\(-1/8g_{9}-63/200g_{8}-7/20g_{7}-21/100g_{6}\)
\(-1/4g_{5}-19/100g_{4}-7/200g_{3}+7/50g_{2}+21/100g_{1}\)
\(1/4h_{5}+19/50h_{4}+7/100h_{3}-7/25h_{2}-21/50h_{1}\)
\(-7/25g_{-1}-21/200g_{-2}+1/50g_{-3}+19/200g_{-4}+3/25g_{-5}\)
\(-7/40g_{-6}-1/8g_{-7}-3/40g_{-8}-1/40g_{-9}\)
\(-1/20g_{-10}-1/20g_{-11}-1/20g_{-12}+1/20g_{-13}\)
\(g_{19}-16/21g_{18}+2/7g_{17}\)
\(5/21g_{16}-10/21g_{15}-2/7g_{14}\)
\(10/21g_{13}-5/21g_{12}+4/21g_{11}+2/7g_{10}\)
\(5/21g_{9}+3/7g_{8}+2/21g_{7}-2/7g_{6}\)
\(10/21g_{5}+4/21g_{4}-1/3g_{3}-8/21g_{2}+2/7g_{1}\)
\(-10/21h_{5}-8/21h_{4}+2/3h_{3}+16/21h_{2}-4/7h_{1}\)
\(-20/21g_{-1}+5/7g_{-2}+10/21g_{-3}-5/21g_{-4}-4/7g_{-5}\)
\(-5/3g_{-6}+5/21g_{-7}+5/7g_{-8}+1/3g_{-9}\)
\(-40/21g_{-10}-10/21g_{-11}+8/21g_{-12}-2/3g_{-13}\)
\(-10/7g_{-14}-6/7g_{-15}+2/7g_{-16}\)
\(-4/7g_{-17}+4/7g_{-18}-4/7g_{-19}\)
\(-g_{23}+9/16g_{22}\)
\(-7/16g_{21}+9/16g_{20}\)
\(-7/8g_{19}+1/8g_{18}+9/16g_{17}\)
\(-3/4g_{16}+11/16g_{15}-9/16g_{14}\)
\(-3/2g_{13}-1/16g_{12}-5/4g_{11}+9/16g_{10}\)
\(-25/16g_{9}-19/16g_{8}+29/16g_{7}-9/16g_{6}\)
\(-25/8g_{5}+3/8g_{4}+3g_{3}-19/8g_{2}+9/16g_{1}\)
\(25/8h_{5}-3/4h_{4}-6h_{3}+19/4h_{2}-9/8h_{1}\)
\(-7/2g_{-1}+133/16g_{-2}-8g_{-3}-7/8g_{-4}+7g_{-5}\)
\(-189/16g_{-6}+261/16g_{-7}-57/8g_{-8}-63/8g_{-9}\)
\(-225/8g_{-10}+375/16g_{-11}+3/4g_{-12}+63/4g_{-13}\)
\(-825/16g_{-14}+363/16g_{-15}-33/2g_{-16}\)
\(-297/4g_{-17}-99/16g_{-18}+33g_{-19}\)
\(1287/16g_{-20}-429/16g_{-21}\)
\(-429/8g_{-22}+429/8g_{-23}\)
\(g_{25}\)
\(g_{24}\)
\(2g_{23}+g_{22}\)
\(3g_{21}+g_{20}\)
\(6g_{19}+4g_{18}+g_{17}\)
\(10g_{16}+5g_{15}-g_{14}\)
\(20g_{13}+15g_{12}-6g_{11}+g_{10}\)
\(35g_{9}-21g_{8}+7g_{7}-g_{6}\)
\(70g_{5}-56g_{4}+28g_{3}-8g_{2}+g_{1}\)
\(-70h_{5}+112h_{4}-56h_{3}+16h_{2}-2h_{1}\)
\(-10g_{-1}+45g_{-2}-120g_{-3}+210g_{-4}-252g_{-5}\)
\(-55g_{-6}+165g_{-7}-330g_{-8}+462g_{-9}\)
\(-220g_{-10}+495g_{-11}-792g_{-12}-924g_{-13}\)
\(-715g_{-14}+1287g_{-15}+1716g_{-16}\)
\(-2002g_{-17}-3003g_{-18}-3432g_{-19}\)
\(5005g_{-20}+6435g_{-21}\)
\(-11440g_{-22}-12870g_{-23}\)
\(24310g_{-24}\)
\(-48620g_{-25}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)		\(18\omega_{1}\)
\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)
\(-18\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)		\(18\omega_{1}\)
\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)
\(-18\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)		\(\displaystyle M_{18\omega_{1}}\oplus M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}
\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}
\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\oplus M_{-18\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)		\(\displaystyle M_{18\omega_{1}}\oplus M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}
\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}
\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\oplus M_{-18\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 56496 arithmetic operations while solving the Serre relations polynomial system.