Type: \(\displaystyle A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1\))
Simple basis: 1 vectors: (1, 1, 1, 1, 1, 1, 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}: D^{1}_6
simple basis centralizer: 6 vectors: (0, 0, 1, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 2), (0, 0, 0, 0, 0, 1, 0)
Number of k-submodules of g: 79
Module decomposition, fundamental coords over k: \(\displaystyle 13V_{2\omega_{1}}+66V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, -1, -2, -2, -2, -2, -2)(0, -1, -2, -2, -2, -2, -2)g_{-47}-\varepsilon_{2}-\varepsilon_{3}
Module 21(0, -1, -1, -2, -2, -2, -2)(0, -1, -1, -2, -2, -2, -2)g_{-45}-\varepsilon_{2}-\varepsilon_{4}
Module 31(0, 0, -1, -2, -2, -2, -2)(0, 0, -1, -2, -2, -2, -2)g_{-43}-\varepsilon_{3}-\varepsilon_{4}
Module 41(0, -1, -1, -1, -2, -2, -2)(0, -1, -1, -1, -2, -2, -2)g_{-42}-\varepsilon_{2}-\varepsilon_{5}
Module 51(0, 0, -1, -1, -2, -2, -2)(0, 0, -1, -1, -2, -2, -2)g_{-40}-\varepsilon_{3}-\varepsilon_{5}
Module 61(0, -1, -1, -1, -1, -2, -2)(0, -1, -1, -1, -1, -2, -2)g_{-39}-\varepsilon_{2}-\varepsilon_{6}
Module 71(0, 0, 0, -1, -2, -2, -2)(0, 0, 0, -1, -2, -2, -2)g_{-37}-\varepsilon_{4}-\varepsilon_{5}
Module 81(0, 0, -1, -1, -1, -2, -2)(0, 0, -1, -1, -1, -2, -2)g_{-36}-\varepsilon_{3}-\varepsilon_{6}
Module 91(0, -1, -1, -1, -1, -1, -2)(0, -1, -1, -1, -1, -1, -2)g_{-35}-\varepsilon_{2}-\varepsilon_{7}
Module 101(0, 0, 0, -1, -1, -2, -2)(0, 0, 0, -1, -1, -2, -2)g_{-33}-\varepsilon_{4}-\varepsilon_{6}
Module 111(0, 0, -1, -1, -1, -1, -2)(0, 0, -1, -1, -1, -1, -2)g_{-32}-\varepsilon_{3}-\varepsilon_{7}
Module 121(0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, -1, -2, -2)g_{-29}-\varepsilon_{5}-\varepsilon_{6}
Module 131(0, 0, 0, -1, -1, -1, -2)(0, 0, 0, -1, -1, -1, -2)g_{-28}-\varepsilon_{4}-\varepsilon_{7}
Module 141(0, -1, -1, -1, -1, -1, 0)(0, -1, -1, -1, -1, -1, 0)g_{-26}-\varepsilon_{2}+\varepsilon_{7}
Module 151(0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, -1, -1, -2)g_{-24}-\varepsilon_{5}-\varepsilon_{7}
Module 161(0, 0, -1, -1, -1, -1, 0)(0, 0, -1, -1, -1, -1, 0)g_{-22}-\varepsilon_{3}+\varepsilon_{7}
Module 171(0, -1, -1, -1, -1, 0, 0)(0, -1, -1, -1, -1, 0, 0)g_{-21}-\varepsilon_{2}+\varepsilon_{6}
Module 181(0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, -1, -2)g_{-19}-\varepsilon_{6}-\varepsilon_{7}
Module 191(0, 0, 0, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, 0)g_{-17}-\varepsilon_{4}+\varepsilon_{7}
Module 201(0, 0, -1, -1, -1, 0, 0)(0, 0, -1, -1, -1, 0, 0)g_{-16}-\varepsilon_{3}+\varepsilon_{6}
Module 211(0, -1, -1, -1, 0, 0, 0)(0, -1, -1, -1, 0, 0, 0)g_{-15}-\varepsilon_{2}+\varepsilon_{5}
Module 221(0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, -1, -1, 0)g_{-12}-\varepsilon_{5}+\varepsilon_{7}
Module 231(0, 0, 0, -1, -1, 0, 0)(0, 0, 0, -1, -1, 0, 0)g_{-11}-\varepsilon_{4}+\varepsilon_{6}
Module 241(0, 0, -1, -1, 0, 0, 0)(0, 0, -1, -1, 0, 0, 0)g_{-10}-\varepsilon_{3}+\varepsilon_{5}
Module 251(0, -1, -1, 0, 0, 0, 0)(0, -1, -1, 0, 0, 0, 0)g_{-9}-\varepsilon_{2}+\varepsilon_{4}
Module 261(0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, -1, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 271(0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, -1, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 281(0, 0, 0, -1, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 291(0, 0, -1, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 301(0, -1, 0, 0, 0, 0, 0)(0, -1, 0, 0, 0, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 313(-1, -2, -2, -2, -2, -2, -2)(1, 0, 0, 0, 0, 0, 0)g_{1}
g_{-31}
g_{-49}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 321(0, 1, 0, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 331(0, 0, 1, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 341(0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 351(0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 1, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 361(0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 1, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 373(-1, -1, -2, -2, -2, -2, -2)(1, 1, 0, 0, 0, 0, 0)g_{8}
g_{-27}
g_{-48}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 381(0, 1, 1, 0, 0, 0, 0)(0, 1, 1, 0, 0, 0, 0)g_{9}\varepsilon_{2}-\varepsilon_{4}
Module 391(0, 0, 1, 1, 0, 0, 0)(0, 0, 1, 1, 0, 0, 0)g_{10}\varepsilon_{3}-\varepsilon_{5}
Module 401(0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 1, 1, 0, 0)g_{11}\varepsilon_{4}-\varepsilon_{6}
Module 411(0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 1, 1, 0)g_{12}\varepsilon_{5}-\varepsilon_{7}
Module 423(-1, -1, -1, -2, -2, -2, -2)(1, 1, 1, 0, 0, 0, 0)g_{14}
g_{-23}
g_{-46}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 431(0, 1, 1, 1, 0, 0, 0)(0, 1, 1, 1, 0, 0, 0)g_{15}\varepsilon_{2}-\varepsilon_{5}
Module 441(0, 0, 1, 1, 1, 0, 0)(0, 0, 1, 1, 1, 0, 0)g_{16}\varepsilon_{3}-\varepsilon_{6}
Module 451(0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 0)g_{17}\varepsilon_{4}-\varepsilon_{7}
Module 461(0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 1, 2)g_{19}\varepsilon_{6}+\varepsilon_{7}
Module 473(-1, -1, -1, -1, -2, -2, -2)(1, 1, 1, 1, 0, 0, 0)g_{20}
g_{-18}
g_{-44}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 481(0, 1, 1, 1, 1, 0, 0)(0, 1, 1, 1, 1, 0, 0)g_{21}\varepsilon_{2}-\varepsilon_{6}
Module 491(0, 0, 1, 1, 1, 1, 0)(0, 0, 1, 1, 1, 1, 0)g_{22}\varepsilon_{3}-\varepsilon_{7}
Module 501(0, 0, 0, 0, 1, 1, 2)(0, 0, 0, 0, 1, 1, 2)g_{24}\varepsilon_{5}+\varepsilon_{7}
Module 513(-1, -1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 1, 0, 0)g_{25}
g_{-13}
g_{-41}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 521(0, 1, 1, 1, 1, 1, 0)(0, 1, 1, 1, 1, 1, 0)g_{26}\varepsilon_{2}-\varepsilon_{7}
Module 531(0, 0, 0, 1, 1, 1, 2)(0, 0, 0, 1, 1, 1, 2)g_{28}\varepsilon_{4}+\varepsilon_{7}
Module 541(0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 1, 2, 2)g_{29}\varepsilon_{5}+\varepsilon_{6}
Module 553(-1, -1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 1, 0)g_{30}
g_{-7}
g_{-38}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 561(0, 0, 1, 1, 1, 1, 2)(0, 0, 1, 1, 1, 1, 2)g_{32}\varepsilon_{3}+\varepsilon_{7}
Module 571(0, 0, 0, 1, 1, 2, 2)(0, 0, 0, 1, 1, 2, 2)g_{33}\varepsilon_{4}+\varepsilon_{6}
Module 583(-1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1)g_{34}
h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-34}		\varepsilon_{1}
0
-\varepsilon_{1}
Module 591(0, 1, 1, 1, 1, 1, 2)(0, 1, 1, 1, 1, 1, 2)g_{35}\varepsilon_{2}+\varepsilon_{7}
Module 601(0, 0, 1, 1, 1, 2, 2)(0, 0, 1, 1, 1, 2, 2)g_{36}\varepsilon_{3}+\varepsilon_{6}
Module 611(0, 0, 0, 1, 2, 2, 2)(0, 0, 0, 1, 2, 2, 2)g_{37}\varepsilon_{4}+\varepsilon_{5}
Module 623(-1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 2)g_{38}
g_{7}
g_{-30}		\varepsilon_{1}+\varepsilon_{7}
\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 631(0, 1, 1, 1, 1, 2, 2)(0, 1, 1, 1, 1, 2, 2)g_{39}\varepsilon_{2}+\varepsilon_{6}
Module 641(0, 0, 1, 1, 2, 2, 2)(0, 0, 1, 1, 2, 2, 2)g_{40}\varepsilon_{3}+\varepsilon_{5}
Module 653(-1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 2, 2)g_{41}
g_{13}
g_{-25}		\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 661(0, 1, 1, 1, 2, 2, 2)(0, 1, 1, 1, 2, 2, 2)g_{42}\varepsilon_{2}+\varepsilon_{5}
Module 671(0, 0, 1, 2, 2, 2, 2)(0, 0, 1, 2, 2, 2, 2)g_{43}\varepsilon_{3}+\varepsilon_{4}
Module 683(-1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 2, 2, 2)g_{44}
g_{18}
g_{-20}		\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 691(0, 1, 1, 2, 2, 2, 2)(0, 1, 1, 2, 2, 2, 2)g_{45}\varepsilon_{2}+\varepsilon_{4}
Module 703(-1, -1, -1, 0, 0, 0, 0)(1, 1, 1, 2, 2, 2, 2)g_{46}
g_{23}
g_{-14}		\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 711(0, 1, 2, 2, 2, 2, 2)(0, 1, 2, 2, 2, 2, 2)g_{47}\varepsilon_{2}+\varepsilon_{3}
Module 723(-1, -1, 0, 0, 0, 0, 0)(1, 1, 2, 2, 2, 2, 2)g_{48}
g_{27}
g_{-8}		\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 733(-1, 0, 0, 0, 0, 0, 0)(1, 2, 2, 2, 2, 2, 2)g_{49}
g_{31}
g_{-1}		\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
Module 741(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{2}0
Module 751(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 761(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 771(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 781(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 791(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{7}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 13
Heirs rejected due to not being maximally dominant: 59
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 59
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^{2}_2, B^{2}_2, 2A^{2}_1, A^{2}_1+A^{1}_1,