Type: \(\displaystyle A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_1\))
Simple basis: 1 vectors: (1, 2, 2, 3, 2, 1)
Simple basis epsilon form:
Simple basis epsilon form with respect to k:
Number of outer autos with trivial action on orthogonal complement and extending to autos of ambient algebra: 0
Number of outer autos with trivial action on orthogonal complement: 0.
C(k_{ss})_{ss}: A^{1}_5
simple basis centralizer: 5 vectors: (1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1), (0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 1, 0, 0)
Number of k-submodules of g: 56
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}}+20V_{\omega_{1}}+35V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(-1, 0, -1, -1, -1, -1)(-1, 0, -1, -1, -1, -1)g_{-24}-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 21(0, 0, -1, -1, -1, -1)(0, 0, -1, -1, -1, -1)g_{-21}-\varepsilon_{1}+\varepsilon_{5}
Module 31(-1, 0, -1, -1, -1, 0)(-1, 0, -1, -1, -1, 0)g_{-18}-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 41(0, 0, 0, -1, -1, -1)(0, 0, 0, -1, -1, -1)g_{-16}-\varepsilon_{2}+\varepsilon_{5}
Module 51(0, 0, -1, -1, -1, 0)(0, 0, -1, -1, -1, 0)g_{-15}-\varepsilon_{1}+\varepsilon_{4}
Module 61(-1, 0, -1, -1, 0, 0)(-1, 0, -1, -1, 0, 0)g_{-12}-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 71(0, 0, 0, 0, -1, -1)(0, 0, 0, 0, -1, -1)g_{-11}-\varepsilon_{3}+\varepsilon_{5}
Module 81(0, 0, 0, -1, -1, 0)(0, 0, 0, -1, -1, 0)g_{-10}-\varepsilon_{2}+\varepsilon_{4}
Module 91(0, 0, -1, -1, 0, 0)(0, 0, -1, -1, 0, 0)g_{-9}-\varepsilon_{1}+\varepsilon_{3}
Module 101(-1, 0, -1, 0, 0, 0)(-1, 0, -1, 0, 0, 0)g_{-7}-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 111(0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, -1)g_{-6}-\varepsilon_{4}+\varepsilon_{5}
Module 121(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{3}+\varepsilon_{4}
Module 131(0, 0, 0, -1, 0, 0)(0, 0, 0, -1, 0, 0)g_{-4}-\varepsilon_{2}+\varepsilon_{3}
Module 141(0, 0, -1, 0, 0, 0)(0, 0, -1, 0, 0, 0)g_{-3}-\varepsilon_{1}+\varepsilon_{2}
Module 151(-1, 0, 0, 0, 0, 0)(-1, 0, 0, 0, 0, 0)g_{-1}1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 161(1, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0)g_{1}-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 172(-1, -1, -2, -3, -2, -1)(0, 1, 0, 0, 0, 0)g_{2}
g_{-35}		-\varepsilon_{1}-\varepsilon_{2}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 181(0, 0, 1, 0, 0, 0)(0, 0, 1, 0, 0, 0)g_{3}\varepsilon_{1}-\varepsilon_{2}
Module 191(0, 0, 0, 1, 0, 0)(0, 0, 0, 1, 0, 0)g_{4}\varepsilon_{2}-\varepsilon_{3}
Module 201(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{3}-\varepsilon_{4}
Module 211(0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 1)g_{6}\varepsilon_{4}-\varepsilon_{5}
Module 221(1, 0, 1, 0, 0, 0)(1, 0, 1, 0, 0, 0)g_{7}1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 232(-1, -1, -2, -2, -2, -1)(0, 1, 0, 1, 0, 0)g_{8}
g_{-34}		-\varepsilon_{1}-\varepsilon_{3}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 241(0, 0, 1, 1, 0, 0)(0, 0, 1, 1, 0, 0)g_{9}\varepsilon_{1}-\varepsilon_{3}
Module 251(0, 0, 0, 1, 1, 0)(0, 0, 0, 1, 1, 0)g_{10}\varepsilon_{2}-\varepsilon_{4}
Module 261(0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 1, 1)g_{11}\varepsilon_{3}-\varepsilon_{5}
Module 271(1, 0, 1, 1, 0, 0)(1, 0, 1, 1, 0, 0)g_{12}1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 282(-1, -1, -1, -2, -2, -1)(0, 1, 1, 1, 0, 0)g_{13}
g_{-33}		-\varepsilon_{2}-\varepsilon_{3}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 292(-1, -1, -2, -2, -1, -1)(0, 1, 0, 1, 1, 0)g_{14}
g_{-32}		-\varepsilon_{1}-\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 301(0, 0, 1, 1, 1, 0)(0, 0, 1, 1, 1, 0)g_{15}\varepsilon_{1}-\varepsilon_{4}
Module 311(0, 0, 0, 1, 1, 1)(0, 0, 0, 1, 1, 1)g_{16}\varepsilon_{2}-\varepsilon_{5}
Module 322(0, -1, -1, -2, -2, -1)(1, 1, 1, 1, 0, 0)g_{17}
g_{-31}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{4}+\varepsilon_{5}
Module 331(1, 0, 1, 1, 1, 0)(1, 0, 1, 1, 1, 0)g_{18}1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 342(-1, -1, -1, -2, -1, -1)(0, 1, 1, 1, 1, 0)g_{19}
g_{-30}		-\varepsilon_{2}-\varepsilon_{4}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 352(-1, -1, -2, -2, -1, 0)(0, 1, 0, 1, 1, 1)g_{20}
g_{-29}		-\varepsilon_{1}-\varepsilon_{5}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 361(0, 0, 1, 1, 1, 1)(0, 0, 1, 1, 1, 1)g_{21}\varepsilon_{1}-\varepsilon_{5}
Module 372(0, -1, -1, -2, -1, -1)(1, 1, 1, 1, 1, 0)g_{22}
g_{-28}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{3}+\varepsilon_{5}
Module 382(-1, -1, -1, -1, -1, -1)(0, 1, 1, 2, 1, 0)g_{23}
g_{-27}		-\varepsilon_{3}-\varepsilon_{4}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 391(1, 0, 1, 1, 1, 1)(1, 0, 1, 1, 1, 1)g_{24}1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 402(-1, -1, -1, -2, -1, 0)(0, 1, 1, 1, 1, 1)g_{25}
g_{-26}		-\varepsilon_{2}-\varepsilon_{5}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 412(0, -1, -1, -1, -1, -1)(1, 1, 1, 2, 1, 0)g_{26}
g_{-25}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{5}
Module 422(0, -1, -1, -2, -1, 0)(1, 1, 1, 1, 1, 1)g_{27}
g_{-23}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{3}+\varepsilon_{4}
Module 432(-1, -1, -1, -1, -1, 0)(0, 1, 1, 2, 1, 1)g_{28}
g_{-22}		-\varepsilon_{3}-\varepsilon_{5}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 442(0, -1, 0, -1, -1, -1)(1, 1, 2, 2, 1, 0)g_{29}
g_{-20}		1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{5}
Module 452(0, -1, -1, -1, -1, 0)(1, 1, 1, 2, 1, 1)g_{30}
g_{-19}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{4}
Module 462(-1, -1, -1, -1, 0, 0)(0, 1, 1, 2, 2, 1)g_{31}
g_{-17}		-\varepsilon_{4}-\varepsilon_{5}
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 472(0, -1, 0, -1, -1, 0)(1, 1, 2, 2, 1, 1)g_{32}
g_{-14}		1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{4}
Module 482(0, -1, -1, -1, 0, 0)(1, 1, 1, 2, 2, 1)g_{33}
g_{-13}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{3}
Module 492(0, -1, 0, -1, 0, 0)(1, 1, 2, 2, 2, 1)g_{34}
g_{-8}		1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{3}
Module 502(0, -1, 0, 0, 0, 0)(1, 1, 2, 3, 2, 1)g_{35}
g_{-2}		1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{2}
Module 513(-1, -2, -2, -3, -2, -1)(1, 2, 2, 3, 2, 1)g_{36}
h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-36}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
0
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 521(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{1}0
Module 531(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{3}0
Module 541(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{4}0
Module 551(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Module 561(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{6}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 1
Heirs rejected due to not being maximally dominant: 48
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 48
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by 0
Potential Dynkin type extensions: A^{1}_2, 2A^{1}_1,