Subalgebra \(A^{156}_1\) ↪ \(E^{1}_6\)
20 out of 119
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: \(\displaystyle E^{1}_6\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{156}_1\): (16, 22, 30, 42, 30, 16): 312
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}\)
Positive simple generators: \(\displaystyle 16g_{6}+30g_{5}+42g_{4}+30g_{3}+22g_{2}+16g_{1}\)
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: \(\displaystyle V_{22\omega_{1}}\oplus V_{16\omega_{1}}\oplus V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{2\omega_{1}}\)
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 6) ; the vectors are over the primal subalgebra.\(g_{6}+15/8g_{5}+21/8g_{4}+15/8g_{3}+11/8g_{2}+g_{1}\)\(g_{21}-11/15g_{20}+g_{18}+11/15g_{17}\)\(-g_{25}+16/11g_{24}-21/8g_{23}+g_{22}\)\(-g_{31}+8/15g_{30}+g_{29}\)\(-g_{33}+g_{32}\)\(g_{36}\)
weight\(2\omega_{1}\)\(8\omega_{1}\)\(10\omega_{1}\)\(14\omega_{1}\)\(16\omega_{1}\)\(22\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{8\omega_{1}} \) → (8)\(\displaystyle V_{10\omega_{1}} \) → (10)\(\displaystyle V_{14\omega_{1}} \) → (14)\(\displaystyle V_{16\omega_{1}} \) → (16)\(\displaystyle V_{22\omega_{1}} \) → (22)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{6}-15/8g_{5}-21/8g_{4}-15/8g_{3}-11/8g_{2}-g_{1}\)
\(h_{6}+15/8h_{5}+21/8h_{4}+15/8h_{3}+11/8h_{2}+h_{1}\)
\(1/8g_{-1}+1/8g_{-2}+1/8g_{-3}+1/8g_{-4}+1/8g_{-5}+1/8g_{-6}\)
\(g_{21}-11/15g_{20}+g_{18}+11/15g_{17}\)
\(4/15g_{16}+11/15g_{14}+11/15g_{13}-4/15g_{12}\)
\(4/15g_{11}+7/15g_{10}+7/15g_{9}+4/15g_{7}\)
\(4/15g_{6}+1/5g_{5}-1/5g_{3}-4/15g_{1}\)
\(-4/15h_{6}-1/5h_{5}+1/5h_{3}+4/15h_{1}\)
\(1/3g_{-1}+2/15g_{-3}-2/15g_{-5}-1/3g_{-6}\)
\(1/5g_{-7}+2/15g_{-9}+2/15g_{-10}+1/5g_{-11}\)
\(1/15g_{-12}-2/15g_{-13}-2/15g_{-14}-1/15g_{-16}\)
\(1/15g_{-17}+1/15g_{-18}-1/15g_{-20}+1/15g_{-21}\)
\(-g_{25}+16/11g_{24}-21/8g_{23}+g_{22}\)
\(5/11g_{21}-g_{20}-5/8g_{19}-5/11g_{18}-g_{17}\)
\(-6/11g_{16}-135/88g_{15}+3/8g_{14}-3/8g_{13}-6/11g_{12}\)
\(-6/11g_{11}-27/44g_{10}+27/44g_{9}-3/4g_{8}+6/11g_{7}\)
\(-6/11g_{6}-3/44g_{5}+21/44g_{4}-3/44g_{3}+3/4g_{2}-6/11g_{1}\)
\(6/11h_{6}+3/44h_{5}-21/44h_{4}+3/44h_{3}-3/4h_{2}+6/11h_{1}\)
\(45/44g_{-1}-45/44g_{-2}+3/44g_{-3}-15/44g_{-4}+3/44g_{-5}+45/44g_{-6}\)
\(21/22g_{-7}-15/22g_{-8}+9/22g_{-9}-9/22g_{-10}-21/22g_{-11}\)
\(6/11g_{-12}+3/11g_{-13}-3/11g_{-14}+9/11g_{-15}+6/11g_{-16}\)
\(-9/11g_{-17}-3/11g_{-18}-3/11g_{-19}-9/11g_{-20}+3/11g_{-21}\)
\(-3/11g_{-22}+3/11g_{-23}-6/11g_{-24}+3/11g_{-25}\)
\(-g_{31}+8/15g_{30}+g_{29}\)
\(-7/15g_{28}+8/15g_{27}+7/15g_{26}\)
\(1/15g_{25}+8/15g_{24}+14/15g_{23}-1/15g_{22}\)
\(3/5g_{21}+1/15g_{20}+4/5g_{19}-3/5g_{18}+1/15g_{17}\)
\(2/3g_{16}-2/5g_{15}+11/15g_{14}-11/15g_{13}+2/3g_{12}\)
\(2/3g_{11}-1/3g_{10}+1/3g_{9}-22/15g_{8}-2/3g_{7}\)
\(2/3g_{6}-g_{5}-4/5g_{4}-g_{3}+22/15g_{2}+2/3g_{1}\)
\(-2/3h_{6}+h_{5}+4/5h_{4}+h_{3}-22/15h_{2}-2/3h_{1}\)
\(-7/3g_{-1}-56/15g_{-2}+28/15g_{-3}+16/15g_{-4}+28/15g_{-5}-7/3g_{-6}\)
\(-21/5g_{-7}-24/5g_{-8}+4/5g_{-9}-4/5g_{-10}+21/5g_{-11}\)
\(-5g_{-12}+4g_{-13}-4g_{-14}+8/5g_{-15}-5g_{-16}\)
\(g_{-17}-33/5g_{-18}+32/5g_{-19}+g_{-20}+33/5g_{-21}\)
\(6/5g_{-22}-32/5g_{-23}-66/5g_{-24}-6/5g_{-25}\)
\(26/5g_{-26}+78/5g_{-27}-26/5g_{-28}\)
\(-26/5g_{-29}-26/5g_{-30}+26/5g_{-31}\)
\(-g_{33}+g_{32}\)
\(-g_{31}-g_{29}\)
\(-g_{28}-g_{26}\)
\(-g_{25}-g_{22}\)
\(-g_{21}-g_{20}-g_{18}+g_{17}\)
\(-2g_{16}+g_{14}+g_{13}+2g_{12}\)
\(-2g_{11}+3g_{10}+3g_{9}-2g_{7}\)
\(-2g_{6}+5g_{5}-5g_{3}+2g_{1}\)
\(2h_{6}-5h_{5}+5h_{3}-2h_{1}\)
\(-9g_{-1}+12g_{-3}-12g_{-5}+9g_{-6}\)
\(-21g_{-7}+12g_{-9}+12g_{-10}-21g_{-11}\)
\(-33g_{-12}-12g_{-13}-12g_{-14}+33g_{-16}\)
\(45g_{-17}-33g_{-18}-45g_{-20}-33g_{-21}\)
\(78g_{-22}+78g_{-25}\)
\(-78g_{-26}-78g_{-28}\)
\(78g_{-29}+78g_{-31}\)
\(78g_{-32}-78g_{-33}\)
\(g_{36}\)
\(g_{35}\)
\(g_{34}\)
\(g_{33}+g_{32}\)
\(g_{31}+2g_{30}-g_{29}\)
\(3g_{28}+2g_{27}-3g_{26}\)
\(5g_{25}+2g_{24}-6g_{23}-5g_{22}\)
\(7g_{21}+5g_{20}-16g_{19}-7g_{18}+5g_{17}\)
\(12g_{16}-30g_{15}-21g_{14}+21g_{13}+12g_{12}\)
\(12g_{11}-63g_{10}+63g_{9}+42g_{8}-12g_{7}\)
\(12g_{6}-75g_{5}+168g_{4}-75g_{3}-42g_{2}+12g_{1}\)
\(-12h_{6}+75h_{5}-168h_{4}+75h_{3}+42h_{2}-12h_{1}\)
\(-99g_{-1}+252g_{-2}+330g_{-3}-528g_{-4}+330g_{-5}-99g_{-6}\)
\(-429g_{-7}+780g_{-8}+858g_{-9}-858g_{-10}+429g_{-11}\)
\(-1287g_{-12}-1638g_{-13}+1638g_{-14}+1716g_{-15}-1287g_{-16}\)
\(2925g_{-17}-3003g_{-18}-4992g_{-19}+2925g_{-20}+3003g_{-21}\)
\(10920g_{-22}+4992g_{-23}-6006g_{-24}-10920g_{-25}\)
\(-15912g_{-26}+27846g_{-27}+15912g_{-28}\)
\(15912g_{-29}-59670g_{-30}-15912g_{-31}\)
\(75582g_{-32}+75582g_{-33}\)
\(-151164g_{-34}\)
\(151164g_{-35}\)
\(-151164g_{-36}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)		\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)		\(22\omega_{1}\)
\(20\omega_{1}\)
\(18\omega_{1}\)
\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)
\(-18\omega_{1}\)
\(-20\omega_{1}\)
\(-22\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)		\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)		\(22\omega_{1}\)
\(20\omega_{1}\)
\(18\omega_{1}\)
\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)
\(-18\omega_{1}\)
\(-20\omega_{1}\)
\(-22\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)		\(\displaystyle M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}
\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}
\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\)		\(\displaystyle M_{22\omega_{1}}\oplus M_{20\omega_{1}}\oplus M_{18\omega_{1}}\oplus M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}
\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\oplus M_{-18\omega_{1}}
\oplus M_{-20\omega_{1}}\oplus M_{-22\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)		\(\displaystyle M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}
\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}
\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\)		\(\displaystyle M_{22\omega_{1}}\oplus M_{20\omega_{1}}\oplus M_{18\omega_{1}}\oplus M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}
\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\oplus M_{-18\omega_{1}}
\oplus M_{-20\omega_{1}}\oplus M_{-22\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 129084 arithmetic operations while solving the Serre relations polynomial system.