Type: \(\displaystyle A^{2}_1+2A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1+2A^{1}_1\))
Simple basis: 3 vectors: (1, 1, 1, 1, 1, 1), (0, 1, 2, 2, 2, 2), (0, 1, 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}: A^{1}_3
simple basis centralizer: 3 vectors: (0, 0, 0, 0, 1, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 2)
Number of k-submodules of g: 31
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}+\omega_{2}+\omega_{3}}+V_{2\omega_{3}}+6V_{\omega_{2}+\omega_{3}}+V_{2\omega_{2}}+7V_{2\omega_{1}}+15V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -1, -2, -2)(0, 0, 0, -1, -2, -2)g_{-24}-\varepsilon_{4}-\varepsilon_{5}
Module 21(0, 0, 0, -1, -1, -2)(0, 0, 0, -1, -1, -2)g_{-20}-\varepsilon_{4}-\varepsilon_{6}
Module 31(0, 0, 0, 0, -1, -2)(0, 0, 0, 0, -1, -2)g_{-16}-\varepsilon_{5}-\varepsilon_{6}
Module 41(0, 0, 0, -1, -1, 0)(0, 0, 0, -1, -1, 0)g_{-10}-\varepsilon_{4}+\varepsilon_{6}
Module 51(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 61(0, 0, 0, -1, 0, 0)(0, 0, 0, -1, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 73(0, -1, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0)g_{2}
h_{2}
g_{-2}		\varepsilon_{2}-\varepsilon_{3}
0
-\varepsilon_{2}+\varepsilon_{3}
Module 81(0, 0, 0, 1, 0, 0)(0, 0, 0, 1, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 91(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 104(0, -1, -1, -2, -2, -2)(0, 1, 1, 0, 0, 0)g_{8}
g_{-30}
g_{3}
g_{-32}		\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 111(0, 0, 0, 1, 1, 0)(0, 0, 0, 1, 1, 0)g_{10}\varepsilon_{4}-\varepsilon_{6}
Module 123(-1, -1, -1, -2, -2, -2)(1, 1, 1, 0, 0, 0)g_{12}
g_{-15}
g_{-33}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 134(0, -1, -1, -1, -2, -2)(0, 1, 1, 1, 0, 0)g_{13}
g_{-27}
g_{9}
g_{-29}		\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 141(0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 1, 2)g_{16}\varepsilon_{5}+\varepsilon_{6}
Module 153(-1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 0, 0)g_{17}
g_{-11}
g_{-31}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 164(0, -1, -1, -1, -1, -2)(0, 1, 1, 1, 1, 0)g_{18}
g_{-23}
g_{14}
g_{-26}		\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
Module 171(0, 0, 0, 1, 1, 2)(0, 0, 0, 1, 1, 2)g_{20}\varepsilon_{4}+\varepsilon_{6}
Module 183(-1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 0)g_{21}
g_{-6}
g_{-28}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 191(0, 0, 0, 1, 2, 2)(0, 0, 0, 1, 2, 2)g_{24}\varepsilon_{4}+\varepsilon_{5}
Module 203(-1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1)g_{25}
h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-25}		\varepsilon_{1}
0
-\varepsilon_{1}
Module 214(0, -1, -1, -1, -1, 0)(0, 1, 1, 1, 1, 2)g_{26}
g_{-14}
g_{23}
g_{-18}		\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
Module 223(-1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 2)g_{28}
g_{6}
g_{-21}		\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 234(0, -1, -1, -1, 0, 0)(0, 1, 1, 1, 2, 2)g_{29}
g_{-9}
g_{27}
g_{-13}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
Module 243(-1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 2, 2)g_{31}
g_{11}
g_{-17}		\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 254(0, -1, -1, 0, 0, 0)(0, 1, 1, 2, 2, 2)g_{32}
g_{-3}
g_{30}
g_{-8}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
Module 263(-1, -1, -1, 0, 0, 0)(1, 1, 1, 2, 2, 2)g_{33}
g_{15}
g_{-12}		\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 273(0, -1, -2, -2, -2, -2)(0, 1, 2, 2, 2, 2)g_{34}
2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}
g_{-34}		\varepsilon_{2}+\varepsilon_{3}
0
-\varepsilon_{2}-\varepsilon_{3}
Module 2812(-1, -2, -2, -2, -2, -2)(1, 2, 2, 2, 2, 2)g_{36}
g_{22}
g_{7}
g_{35}
g_{-1}
g_{-19}
g_{19}
g_{1}
g_{-35}
g_{-7}
g_{-22}
g_{-36}		\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
-\varepsilon_{3}
\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 291(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{4}0
Module 301(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Module 311(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{6}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 16
Heirs rejected due to not being maximally dominant: 11
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 11
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{2}_1+A^{1}_1
Potential Dynkin type extensions: A^{2}_1+3A^{1}_1,