Type: \(\displaystyle 3A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 3A^{1}_1\))
Simple basis: 3 vectors: (1, 2, 2, 2, 2, 2, 2, 2), (1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 1, 2, 2, 2, 2, 2)
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}: B^{1}_4+A^{1}_1
simple basis centralizer: 5 vectors: (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of k-submodules of g: 71
Module decomposition, fundamental coords over k: \(\displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}}+V_{2\omega_{3}}+V_{2\omega_{2}}+9V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+18V_{\omega_{3}}+39V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, -1, -2, -2, -2)(0, 0, 0, 0, -1, -2, -2, -2)g_{-44}-\varepsilon_{5}-\varepsilon_{6}
Module 21(0, 0, 0, 0, -1, -1, -2, -2)(0, 0, 0, 0, -1, -1, -2, -2)g_{-39}-\varepsilon_{5}-\varepsilon_{7}
Module 31(0, 0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, 0, -1, -2, -2)g_{-34}-\varepsilon_{6}-\varepsilon_{7}
Module 41(0, 0, 0, 0, -1, -1, -1, -2)(0, 0, 0, 0, -1, -1, -1, -2)g_{-33}-\varepsilon_{5}-\varepsilon_{8}
Module 51(0, 0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, 0, -1, -1, -2)g_{-28}-\varepsilon_{6}-\varepsilon_{8}
Module 61(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, -1, -1, -1, -1)g_{-27}-\varepsilon_{5}
Module 71(0, 0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, 0, -1, -2)g_{-22}-\varepsilon_{7}-\varepsilon_{8}
Module 81(0, 0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, 0, -1, -1, -1)g_{-21}-\varepsilon_{6}
Module 91(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, -1, -1, -1, 0)g_{-20}-\varepsilon_{5}+\varepsilon_{8}
Module 101(0, 0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, 0, -1, -1)g_{-15}-\varepsilon_{7}
Module 111(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, -1, -1, 0)g_{-14}-\varepsilon_{6}+\varepsilon_{8}
Module 121(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, -1, -1, 0, 0)g_{-13}-\varepsilon_{5}+\varepsilon_{7}
Module 131(0, 0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, 0, -1)g_{-8}-\varepsilon_{8}
Module 141(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 151(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 161(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 171(0, 0, -1, 0, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 183(-1, 0, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}
h_{1}
g_{-1}		\varepsilon_{1}-\varepsilon_{2}
0
-\varepsilon_{1}+\varepsilon_{2}
Module 191(0, 0, 1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 202(0, 0, -1, -1, -2, -2, -2, -2)(0, 0, 0, 1, 0, 0, 0, 0)g_{4}
g_{-55}		\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 211(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 221(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 231(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 241(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}\varepsilon_{8}
Module 252(0, 0, 0, -1, -2, -2, -2, -2)(0, 0, 1, 1, 0, 0, 0, 0)g_{11}
g_{-52}		\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 262(0, 0, -1, -1, -1, -2, -2, -2)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}
g_{-51}		\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{3}-\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 281(0, 0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}\varepsilon_{6}-\varepsilon_{8}
Module 291(0, 0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 0, 1, 1)g_{15}\varepsilon_{7}
Module 302(0, 0, 0, -1, -1, -2, -2, -2)(0, 0, 1, 1, 1, 0, 0, 0)g_{18}
g_{-48}		\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
Module 312(0, 0, -1, -1, -1, -1, -2, -2)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}
g_{-47}		\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 321(0, 0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}\varepsilon_{5}-\varepsilon_{8}
Module 331(0, 0, 0, 0, 0, 1, 1, 1)(0, 0, 0, 0, 0, 1, 1, 1)g_{21}\varepsilon_{6}
Module 341(0, 0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 0, 1, 2)g_{22}\varepsilon_{7}+\varepsilon_{8}
Module 354(-1, -1, -1, -1, -2, -2, -2, -2)(1, 1, 1, 1, 0, 0, 0, 0)g_{23}
g_{-57}
g_{17}
g_{-59}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 362(0, 0, 0, -1, -1, -1, -2, -2)(0, 0, 1, 1, 1, 1, 0, 0)g_{25}
g_{-43}		\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
Module 372(0, 0, -1, -1, -1, -1, -1, -2)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}
g_{-42}		\varepsilon_{4}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
Module 381(0, 0, 0, 0, 1, 1, 1, 1)(0, 0, 0, 0, 1, 1, 1, 1)g_{27}\varepsilon_{5}
Module 391(0, 0, 0, 0, 0, 1, 1, 2)(0, 0, 0, 0, 0, 1, 1, 2)g_{28}\varepsilon_{6}+\varepsilon_{8}
Module 404(-1, -1, -1, -1, -1, -2, -2, -2)(1, 1, 1, 1, 1, 0, 0, 0)g_{29}
g_{-54}
g_{24}
g_{-56}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 412(0, 0, 0, -1, -1, -1, -1, -2)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}
g_{-38}		\varepsilon_{3}-\varepsilon_{8}
-\varepsilon_{4}-\varepsilon_{8}
Module 422(0, 0, -1, -1, -1, -1, -1, -1)(0, 0, 0, 1, 1, 1, 1, 1)g_{32}
g_{-37}		\varepsilon_{4}
-\varepsilon_{3}
Module 431(0, 0, 0, 0, 1, 1, 1, 2)(0, 0, 0, 0, 1, 1, 1, 2)g_{33}\varepsilon_{5}+\varepsilon_{8}
Module 441(0, 0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 0, 1, 2, 2)g_{34}\varepsilon_{6}+\varepsilon_{7}
Module 454(-1, -1, -1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 1, 1, 0, 0)g_{35}
g_{-50}
g_{30}
g_{-53}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 462(0, 0, 0, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 1, 1)g_{37}
g_{-32}		\varepsilon_{3}
-\varepsilon_{4}
Module 472(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, 0, 1, 1, 1, 1, 2)g_{38}
g_{-31}		\varepsilon_{4}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
Module 481(0, 0, 0, 0, 1, 1, 2, 2)(0, 0, 0, 0, 1, 1, 2, 2)g_{39}\varepsilon_{5}+\varepsilon_{7}
Module 494(-1, -1, -1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-46}
g_{36}
g_{-49}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
Module 502(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 1, 2)g_{42}
g_{-26}		\varepsilon_{3}+\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{8}
Module 512(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, 0, 1, 1, 1, 2, 2)g_{43}
g_{-25}		\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 521(0, 0, 0, 0, 1, 2, 2, 2)(0, 0, 0, 0, 1, 2, 2, 2)g_{44}\varepsilon_{5}+\varepsilon_{6}
Module 534(-1, -1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1, 1)g_{45}
g_{-41}
g_{41}
g_{-45}		\varepsilon_{1}
-\varepsilon_{2}
\varepsilon_{2}
-\varepsilon_{1}
Module 542(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 1, 2, 2)g_{47}
g_{-19}		\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
Module 552(0, 0, -1, -1, -1, 0, 0, 0)(0, 0, 0, 1, 1, 2, 2, 2)g_{48}
g_{-18}		\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
Module 564(-1, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 2)g_{49}
g_{-36}
g_{46}
g_{-40}		\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 572(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 1, 1, 1, 2, 2, 2)g_{51}
g_{-12}		\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
Module 582(0, 0, -1, -1, 0, 0, 0, 0)(0, 0, 0, 1, 2, 2, 2, 2)g_{52}
g_{-11}		\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
Module 594(-1, -1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 1, 2, 2)g_{53}
g_{-30}
g_{50}
g_{-35}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 602(0, 0, 0, -1, 0, 0, 0, 0)(0, 0, 1, 1, 2, 2, 2, 2)g_{55}
g_{-4}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
Module 614(-1, -1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 1, 2, 2, 2)g_{56}
g_{-24}
g_{54}
g_{-29}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 623(0, 0, -1, -2, -2, -2, -2, -2)(0, 0, 1, 2, 2, 2, 2, 2)g_{58}
2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}
g_{-58}		\varepsilon_{3}+\varepsilon_{4}
0
-\varepsilon_{3}-\varepsilon_{4}
Module 634(-1, -1, -1, -1, 0, 0, 0, 0)(1, 1, 1, 1, 2, 2, 2, 2)g_{59}
g_{-17}
g_{57}
g_{-23}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 648(-1, -1, -2, -2, -2, -2, -2, -2)(1, 1, 1, 2, 2, 2, 2, 2)g_{61}
g_{-10}
g_{60}
g_{9}
g_{-16}
g_{-62}
g_{2}
g_{-63}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 658(-1, -1, -1, -2, -2, -2, -2, -2)(1, 1, 2, 2, 2, 2, 2, 2)g_{63}
g_{-2}
g_{62}
g_{16}
g_{-9}
g_{-60}
g_{10}
g_{-61}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 663(-1, -2, -2, -2, -2, -2, -2, -2)(1, 2, 2, 2, 2, 2, 2, 2)g_{64}
2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-64}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 671(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 681(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 691(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 701(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 711(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: 40
Heirs rejected due to not being maximally dominant: 24
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 24
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by 2A^{1}_1
Potential Dynkin type extensions: 4A^{1}_1,