Type: \(\displaystyle A^{2}_2\) (Dynkin type computed to be: \(\displaystyle A^{2}_2\))
Simple basis: 2 vectors: (1, 2, 2, 2, 2, 2, 2, 1), (0, -1, 0, 0, 0, 0, 0, 0)
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}: C^{1}_5
simple basis centralizer: 5 vectors: (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, 1, 0, 0)
Number of k-submodules of g: 79
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+10V_{\omega_{2}}+10V_{\omega_{1}}+56V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -2, -2, -2, -2, -1)(0, 0, 0, -2, -2, -2, -2, -1)g_{-52}-2\varepsilon_{4}
Module 21(0, 0, 0, -1, -2, -2, -2, -1)(0, 0, 0, -1, -2, -2, -2, -1)g_{-48}-\varepsilon_{4}-\varepsilon_{5}
Module 31(0, 0, 0, 0, -2, -2, -2, -1)(0, 0, 0, 0, -2, -2, -2, -1)g_{-44}-2\varepsilon_{5}
Module 41(0, 0, 0, -1, -1, -2, -2, -1)(0, 0, 0, -1, -1, -2, -2, -1)g_{-43}-\varepsilon_{4}-\varepsilon_{6}
Module 51(0, 0, 0, 0, -1, -2, -2, -1)(0, 0, 0, 0, -1, -2, -2, -1)g_{-39}-\varepsilon_{5}-\varepsilon_{6}
Module 61(0, 0, 0, -1, -1, -1, -2, -1)(0, 0, 0, -1, -1, -1, -2, -1)g_{-38}-\varepsilon_{4}-\varepsilon_{7}
Module 71(0, 0, 0, 0, 0, -2, -2, -1)(0, 0, 0, 0, 0, -2, -2, -1)g_{-34}-2\varepsilon_{6}
Module 81(0, 0, 0, 0, -1, -1, -2, -1)(0, 0, 0, 0, -1, -1, -2, -1)g_{-33}-\varepsilon_{5}-\varepsilon_{7}
Module 91(0, 0, 0, -1, -1, -1, -1, -1)(0, 0, 0, -1, -1, -1, -1, -1)g_{-32}-\varepsilon_{4}-\varepsilon_{8}
Module 101(0, 0, 0, 0, 0, -1, -2, -1)(0, 0, 0, 0, 0, -1, -2, -1)g_{-28}-\varepsilon_{6}-\varepsilon_{7}
Module 111(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, -1, -1, -1, -1)g_{-27}-\varepsilon_{5}-\varepsilon_{8}
Module 121(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, -1, 0)g_{-26}-\varepsilon_{4}+\varepsilon_{8}
Module 131(0, 0, 0, 0, 0, 0, -2, -1)(0, 0, 0, 0, 0, 0, -2, -1)g_{-22}-2\varepsilon_{7}
Module 141(0, 0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, 0, -1, -1, -1)g_{-21}-\varepsilon_{6}-\varepsilon_{8}
Module 151(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, -1, -1, -1, 0)g_{-20}-\varepsilon_{5}+\varepsilon_{8}
Module 161(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 0, -1, -1, -1, 0, 0)g_{-19}-\varepsilon_{4}+\varepsilon_{7}
Module 171(0, 0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, 0, -1, -1)g_{-15}-\varepsilon_{7}-\varepsilon_{8}
Module 181(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, -1, -1, 0)g_{-14}-\varepsilon_{6}+\varepsilon_{8}
Module 191(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, -1, -1, 0, 0)g_{-13}-\varepsilon_{5}+\varepsilon_{7}
Module 201(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 0, -1, -1, 0, 0, 0)g_{-12}-\varepsilon_{4}+\varepsilon_{6}
Module 211(0, 0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, 0, -1)g_{-8}-2\varepsilon_{8}
Module 221(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 231(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 241(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 251(0, 0, 0, -1, 0, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 263(-1, -1, -1, -2, -2, -2, -2, -1)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}
g_{10}
g_{-59}		\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 271(0, 0, 0, 1, 0, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 281(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 291(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 301(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 311(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}2\varepsilon_{8}
Module 323(-1, -1, -1, -1, -2, -2, -2, -1)(0, 0, 1, 1, 0, 0, 0, 0)g_{11}
g_{17}
g_{-56}		\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 331(0, 0, 0, 1, 1, 0, 0, 0)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}\varepsilon_{4}-\varepsilon_{6}
Module 341(0, 0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}\varepsilon_{5}-\varepsilon_{7}
Module 351(0, 0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}\varepsilon_{6}-\varepsilon_{8}
Module 361(0, 0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 0, 1, 1)g_{15}\varepsilon_{7}+\varepsilon_{8}
Module 373(0, 0, -1, -2, -2, -2, -2, -1)(1, 1, 1, 0, 0, 0, 0, 0)g_{16}
g_{-57}
g_{-55}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 383(-1, -1, -1, -1, -1, -2, -2, -1)(0, 0, 1, 1, 1, 0, 0, 0)g_{18}
g_{24}
g_{-53}		\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 391(0, 0, 0, 1, 1, 1, 0, 0)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}\varepsilon_{4}-\varepsilon_{7}
Module 401(0, 0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}\varepsilon_{5}-\varepsilon_{8}
Module 411(0, 0, 0, 0, 0, 1, 1, 1)(0, 0, 0, 0, 0, 1, 1, 1)g_{21}\varepsilon_{6}+\varepsilon_{8}
Module 421(0, 0, 0, 0, 0, 0, 2, 1)(0, 0, 0, 0, 0, 0, 2, 1)g_{22}2\varepsilon_{7}
Module 433(0, 0, -1, -1, -2, -2, -2, -1)(1, 1, 1, 1, 0, 0, 0, 0)g_{23}
g_{-54}
g_{-51}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 443(-1, -1, -1, -1, -1, -1, -2, -1)(0, 0, 1, 1, 1, 1, 0, 0)g_{25}
g_{30}
g_{-49}		\varepsilon_{3}-\varepsilon_{7}
\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 451(0, 0, 0, 1, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}\varepsilon_{4}-\varepsilon_{8}
Module 461(0, 0, 0, 0, 1, 1, 1, 1)(0, 0, 0, 0, 1, 1, 1, 1)g_{27}\varepsilon_{5}+\varepsilon_{8}
Module 471(0, 0, 0, 0, 0, 1, 2, 1)(0, 0, 0, 0, 0, 1, 2, 1)g_{28}\varepsilon_{6}+\varepsilon_{7}
Module 483(0, 0, -1, -1, -1, -2, -2, -1)(1, 1, 1, 1, 1, 0, 0, 0)g_{29}
g_{-50}
g_{-47}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 493(-1, -1, -1, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}
g_{36}
g_{-45}		\varepsilon_{3}-\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
Module 501(0, 0, 0, 1, 1, 1, 1, 1)(0, 0, 0, 1, 1, 1, 1, 1)g_{32}\varepsilon_{4}+\varepsilon_{8}
Module 511(0, 0, 0, 0, 1, 1, 2, 1)(0, 0, 0, 0, 1, 1, 2, 1)g_{33}\varepsilon_{5}+\varepsilon_{7}
Module 521(0, 0, 0, 0, 0, 2, 2, 1)(0, 0, 0, 0, 0, 2, 2, 1)g_{34}2\varepsilon_{6}
Module 533(0, 0, -1, -1, -1, -1, -2, -1)(1, 1, 1, 1, 1, 1, 0, 0)g_{35}
g_{-46}
g_{-42}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 543(-1, -1, -1, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 1, 1)g_{37}
g_{41}
g_{-40}		\varepsilon_{3}+\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 551(0, 0, 0, 1, 1, 1, 2, 1)(0, 0, 0, 1, 1, 1, 2, 1)g_{38}\varepsilon_{4}+\varepsilon_{7}
Module 561(0, 0, 0, 0, 1, 2, 2, 1)(0, 0, 0, 0, 1, 2, 2, 1)g_{39}\varepsilon_{5}+\varepsilon_{6}
Module 573(0, 0, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-41}
g_{-37}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
Module 583(-1, -1, -1, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 1, 2, 1)g_{42}
g_{46}
g_{-35}		\varepsilon_{3}+\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 591(0, 0, 0, 1, 1, 2, 2, 1)(0, 0, 0, 1, 1, 2, 2, 1)g_{43}\varepsilon_{4}+\varepsilon_{6}
Module 601(0, 0, 0, 0, 2, 2, 2, 1)(0, 0, 0, 0, 2, 2, 2, 1)g_{44}2\varepsilon_{5}
Module 613(0, 0, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 1)g_{45}
g_{-36}
g_{-31}		\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
Module 623(-1, -1, -1, -1, -1, 0, 0, 0)(0, 0, 1, 1, 1, 2, 2, 1)g_{47}
g_{50}
g_{-29}		\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 631(0, 0, 0, 1, 2, 2, 2, 1)(0, 0, 0, 1, 2, 2, 2, 1)g_{48}\varepsilon_{4}+\varepsilon_{5}
Module 643(0, 0, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 1, 2, 1)g_{49}
g_{-30}
g_{-25}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 653(-1, -1, -1, -1, 0, 0, 0, 0)(0, 0, 1, 1, 2, 2, 2, 1)g_{51}
g_{54}
g_{-23}		\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 661(0, 0, 0, 2, 2, 2, 2, 1)(0, 0, 0, 2, 2, 2, 2, 1)g_{52}2\varepsilon_{4}
Module 673(0, 0, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 1, 2, 2, 1)g_{53}
g_{-24}
g_{-18}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
Module 683(-1, -1, -1, 0, 0, 0, 0, 0)(0, 0, 1, 2, 2, 2, 2, 1)g_{55}
g_{57}
g_{-16}		\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 693(0, 0, -1, -1, 0, 0, 0, 0)(1, 1, 1, 1, 2, 2, 2, 1)g_{56}
g_{-17}
g_{-11}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
Module 706(-2, -2, -2, -2, -2, -2, -2, -1)(0, 0, 2, 2, 2, 2, 2, 1)g_{58}
g_{60}
g_{-9}
g_{62}
g_{-1}
g_{-64}		2\varepsilon_{3}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
2\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
-2\varepsilon_{1}
Module 713(0, 0, -1, 0, 0, 0, 0, 0)(1, 1, 1, 2, 2, 2, 2, 1)g_{59}
g_{-10}
g_{-3}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
Module 728(-1, -1, -2, -2, -2, -2, -2, -1)(1, 1, 2, 2, 2, 2, 2, 1)g_{61}
g_{-2}
g_{63}
-h_{2}
h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-63}
g_{2}
g_{-61}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{2}
0
0
-\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 736(0, 0, -2, -2, -2, -2, -2, -1)(2, 2, 2, 2, 2, 2, 2, 1)g_{64}
g_{1}
g_{-62}
g_{9}
g_{-60}
g_{-58}		2\varepsilon_{1}
\varepsilon_{1}-\varepsilon_{2}
-2\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
-2\varepsilon_{3}
Module 741(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{3}-h_{1}0
Module 751(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 761(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 771(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 781(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 791(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{8}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 12
Heirs rejected due to not being maximally dominant: 57
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 57
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{2}_1
Potential Dynkin type extensions: A^{2}_3, B^{2}_3, C^{1}_3, A^{2}_2+A^{2}_1, A^{2}_2+A^{1}_1,