Type: \(\displaystyle A^{1}_2\) (Dynkin type computed to be: \(\displaystyle A^{1}_2\))
Simple basis: 2 vectors: (1, 1, 1, 1, 1, 1, 1, 1), (0, 0, 0, 0, 0, 0, 0, -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: (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0)
Number of k-submodules of g: 49
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{2}}+6V_{\omega_{2}}+6V_{\omega_{1}}+36V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, -1, -1, -1, -1, -1, 0, 0)(0, -1, -1, -1, -1, -1, 0, 0)g_{-28}-\varepsilon_{2}+\varepsilon_{7}
Module 21(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, -1, -1, -1, -1, 0, 0)g_{-24}-\varepsilon_{3}+\varepsilon_{7}
Module 31(0, -1, -1, -1, -1, 0, 0, 0)(0, -1, -1, -1, -1, 0, 0, 0)g_{-23}-\varepsilon_{2}+\varepsilon_{6}
Module 41(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 0, -1, -1, -1, 0, 0)g_{-19}-\varepsilon_{4}+\varepsilon_{7}
Module 51(0, 0, -1, -1, -1, 0, 0, 0)(0, 0, -1, -1, -1, 0, 0, 0)g_{-18}-\varepsilon_{3}+\varepsilon_{6}
Module 61(0, -1, -1, -1, 0, 0, 0, 0)(0, -1, -1, -1, 0, 0, 0, 0)g_{-17}-\varepsilon_{2}+\varepsilon_{5}
Module 71(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, -1, -1, 0, 0)g_{-13}-\varepsilon_{5}+\varepsilon_{7}
Module 81(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 0, -1, -1, 0, 0, 0)g_{-12}-\varepsilon_{4}+\varepsilon_{6}
Module 91(0, 0, -1, -1, 0, 0, 0, 0)(0, 0, -1, -1, 0, 0, 0, 0)g_{-11}-\varepsilon_{3}+\varepsilon_{5}
Module 101(0, -1, -1, 0, 0, 0, 0, 0)(0, -1, -1, 0, 0, 0, 0, 0)g_{-10}-\varepsilon_{2}+\varepsilon_{4}
Module 111(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 121(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 131(0, 0, 0, -1, 0, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 141(0, 0, -1, 0, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 151(0, -1, 0, 0, 0, 0, 0, 0)(0, -1, 0, 0, 0, 0, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 163(0, -1, -1, -1, -1, -1, -1, 0)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}
g_{-35}
g_{-32}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{9}
-\varepsilon_{2}+\varepsilon_{8}
Module 171(0, 1, 0, 0, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 181(0, 0, 1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 191(0, 0, 0, 1, 0, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 201(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 211(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 223(-1, -1, -1, -1, -1, -1, 0, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}
g_{15}
g_{-31}		\varepsilon_{7}-\varepsilon_{8}
\varepsilon_{7}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{7}
Module 233(0, 0, -1, -1, -1, -1, -1, 0)(1, 1, 0, 0, 0, 0, 0, 0)g_{9}
g_{-33}
g_{-29}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{9}
-\varepsilon_{3}+\varepsilon_{8}
Module 241(0, 1, 1, 0, 0, 0, 0, 0)(0, 1, 1, 0, 0, 0, 0, 0)g_{10}\varepsilon_{2}-\varepsilon_{4}
Module 251(0, 0, 1, 1, 0, 0, 0, 0)(0, 0, 1, 1, 0, 0, 0, 0)g_{11}\varepsilon_{3}-\varepsilon_{5}
Module 261(0, 0, 0, 1, 1, 0, 0, 0)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}\varepsilon_{4}-\varepsilon_{6}
Module 271(0, 0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}\varepsilon_{5}-\varepsilon_{7}
Module 283(-1, -1, -1, -1, -1, 0, 0, 0)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}
g_{21}
g_{-27}		\varepsilon_{6}-\varepsilon_{8}
\varepsilon_{6}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{6}
Module 293(0, 0, 0, -1, -1, -1, -1, 0)(1, 1, 1, 0, 0, 0, 0, 0)g_{16}
g_{-30}
g_{-25}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}+\varepsilon_{9}
-\varepsilon_{4}+\varepsilon_{8}
Module 301(0, 1, 1, 1, 0, 0, 0, 0)(0, 1, 1, 1, 0, 0, 0, 0)g_{17}\varepsilon_{2}-\varepsilon_{5}
Module 311(0, 0, 1, 1, 1, 0, 0, 0)(0, 0, 1, 1, 1, 0, 0, 0)g_{18}\varepsilon_{3}-\varepsilon_{6}
Module 321(0, 0, 0, 1, 1, 1, 0, 0)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}\varepsilon_{4}-\varepsilon_{7}
Module 333(-1, -1, -1, -1, 0, 0, 0, 0)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}
g_{26}
g_{-22}		\varepsilon_{5}-\varepsilon_{8}
\varepsilon_{5}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{5}
Module 343(0, 0, 0, 0, -1, -1, -1, 0)(1, 1, 1, 1, 0, 0, 0, 0)g_{22}
g_{-26}
g_{-20}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{5}+\varepsilon_{9}
-\varepsilon_{5}+\varepsilon_{8}
Module 351(0, 1, 1, 1, 1, 0, 0, 0)(0, 1, 1, 1, 1, 0, 0, 0)g_{23}\varepsilon_{2}-\varepsilon_{6}
Module 361(0, 0, 1, 1, 1, 1, 0, 0)(0, 0, 1, 1, 1, 1, 0, 0)g_{24}\varepsilon_{3}-\varepsilon_{7}
Module 373(-1, -1, -1, 0, 0, 0, 0, 0)(0, 0, 0, 1, 1, 1, 1, 0)g_{25}
g_{30}
g_{-16}		\varepsilon_{4}-\varepsilon_{8}
\varepsilon_{4}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{4}
Module 383(0, 0, 0, 0, 0, -1, -1, 0)(1, 1, 1, 1, 1, 0, 0, 0)g_{27}
g_{-21}
g_{-14}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{6}+\varepsilon_{9}
-\varepsilon_{6}+\varepsilon_{8}
Module 391(0, 1, 1, 1, 1, 1, 0, 0)(0, 1, 1, 1, 1, 1, 0, 0)g_{28}\varepsilon_{2}-\varepsilon_{7}
Module 403(-1, -1, 0, 0, 0, 0, 0, 0)(0, 0, 1, 1, 1, 1, 1, 0)g_{29}
g_{33}
g_{-9}		\varepsilon_{3}-\varepsilon_{8}
\varepsilon_{3}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{3}
Module 413(0, 0, 0, 0, 0, 0, -1, 0)(1, 1, 1, 1, 1, 1, 0, 0)g_{31}
g_{-15}
g_{-7}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{7}+\varepsilon_{9}
-\varepsilon_{7}+\varepsilon_{8}
Module 423(-1, 0, 0, 0, 0, 0, 0, 0)(0, 1, 1, 1, 1, 1, 1, 0)g_{32}
g_{35}
g_{-1}		\varepsilon_{2}-\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{2}
Module 438(-1, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 0)g_{34}
g_{-8}
g_{36}
-h_{8}
h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-36}
g_{8}
g_{-34}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{8}+\varepsilon_{9}
\varepsilon_{1}-\varepsilon_{9}
0
0
-\varepsilon_{1}+\varepsilon_{9}
\varepsilon_{8}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{8}
Module 441(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{2}0
Module 451(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 461(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 471(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 481(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 491(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{8}+2h_{7}-h_{1}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 7
Heirs rejected due to not being maximally dominant: 34
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 34
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{1}_1
Potential Dynkin type extensions: A^{1}_3, A^{1}_2+A^{1}_1,