Type: \(\displaystyle 3A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 3A^{1}_1\))
Simple basis: 3 vectors: (1, 2, 2, 2, 2, 2), (0, 0, 1, 2, 2, 2), (0, 0, 0, 0, 1, 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}: 3A^{1}_1
simple basis centralizer: 3 vectors: (0, 0, 0, 0, 1, 0), (0, 0, 1, 0, 0, 0), (1, 0, 0, 0, 0, 0)
Number of k-submodules of g: 30
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{3}}+4V_{\omega_{2}+\omega_{3}}+4V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+4V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+2V_{\omega_{3}}+2V_{\omega_{2}}+2V_{\omega_{1}}+9V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 21(0, 0, -1, 0, 0, 0)(0, 0, -1, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 31(-1, 0, 0, 0, 0, 0)(-1, 0, 0, 0, 0, 0)g_{-1}-\varepsilon_{1}+\varepsilon_{2}
Module 41(1, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0)g_{1}\varepsilon_{1}-\varepsilon_{2}
Module 51(0, 0, 1, 0, 0, 0)(0, 0, 1, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 61(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 72(0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, 1)g_{6}
g_{-11}		\varepsilon_{6}
-\varepsilon_{5}
Module 82(0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 1, 1)g_{11}
g_{-6}		\varepsilon_{5}
-\varepsilon_{6}
Module 92(0, 0, -1, -1, -1, -1)(0, 0, 0, 1, 1, 1)g_{15}
g_{-19}		\varepsilon_{4}
-\varepsilon_{3}
Module 103(0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 1, 2)g_{16}
2h_{6}+h_{5}
g_{-16}		\varepsilon_{5}+\varepsilon_{6}
0
-\varepsilon_{5}-\varepsilon_{6}
Module 112(0, 0, 0, -1, -1, -1)(0, 0, 1, 1, 1, 1)g_{19}
g_{-15}		\varepsilon_{3}
-\varepsilon_{4}
Module 124(0, 0, -1, -1, -2, -2)(0, 0, 0, 1, 1, 2)g_{20}
g_{-14}
g_{4}
g_{-27}		\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 132(-1, -1, -1, -1, -1, -1)(0, 1, 1, 1, 1, 1)g_{22}
g_{-25}		\varepsilon_{2}
-\varepsilon_{1}
Module 144(0, 0, 0, -1, -2, -2)(0, 0, 1, 1, 1, 2)g_{23}
g_{-10}
g_{9}
g_{-24}		\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 154(0, 0, -1, -1, -1, -2)(0, 0, 0, 1, 2, 2)g_{24}
g_{-9}
g_{10}
g_{-23}		\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 162(0, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1)g_{25}
g_{-22}		\varepsilon_{1}
-\varepsilon_{2}
Module 174(-1, -1, -1, -1, -2, -2)(0, 1, 1, 1, 1, 2)g_{26}
g_{-21}
g_{13}
g_{-31}		\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 184(0, 0, 0, -1, -1, -2)(0, 0, 1, 1, 2, 2)g_{27}
g_{-4}
g_{14}
g_{-20}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
Module 194(0, -1, -1, -1, -2, -2)(1, 1, 1, 1, 1, 2)g_{28}
g_{-18}
g_{17}
g_{-29}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 204(-1, -1, -1, -1, -1, -2)(0, 1, 1, 1, 2, 2)g_{29}
g_{-17}
g_{18}
g_{-28}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 213(0, 0, -1, -2, -2, -2)(0, 0, 1, 2, 2, 2)g_{30}
2h_{6}+2h_{5}+2h_{4}+h_{3}
g_{-30}		\varepsilon_{3}+\varepsilon_{4}
0
-\varepsilon_{3}-\varepsilon_{4}
Module 224(0, -1, -1, -1, -1, -2)(1, 1, 1, 1, 2, 2)g_{31}
g_{-13}
g_{21}
g_{-26}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
Module 234(-1, -1, -2, -2, -2, -2)(0, 1, 1, 2, 2, 2)g_{32}
g_{-12}
g_{2}
g_{-35}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 244(0, -1, -2, -2, -2, -2)(1, 1, 1, 2, 2, 2)g_{33}
g_{-8}
g_{7}
g_{-34}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 254(-1, -1, -1, -2, -2, -2)(0, 1, 2, 2, 2, 2)g_{34}
g_{-7}
g_{8}
g_{-33}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 264(0, -1, -1, -2, -2, -2)(1, 1, 2, 2, 2, 2)g_{35}
g_{-2}
g_{12}
g_{-32}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 273(-1, -2, -2, -2, -2, -2)(1, 2, 2, 2, 2, 2)g_{36}
2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-36}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 281(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{1}0
Module 291(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{3}0
Module 301(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 21
Heirs rejected due to not being maximally dominant: 3
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 3
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 3
Parabolically induced by 2A^{1}_1
Potential Dynkin type extensions: 4A^{1}_1,