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

Semisimple subalgebra: W_{1}
Centralizer extension: 0


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