Type: \(\displaystyle 3A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 3A^{1}_1\))
Simple basis: 3 vectors: (1, 2, 2, 3, 2, 1), (1, 0, 1, 1, 1, 1), (0, 0, 1, 1, 1, 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}_1
simple basis centralizer: 1 vectors: (0, 0, 0, 1, 0, 0)
Number of k-submodules of g: 28
Module decomposition, fundamental coords over k: \(\displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}}+V_{2\omega_{3}}+2V_{\omega_{2}+\omega_{3}}+2V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+2V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+4V_{\omega_{3}}+4V_{\omega_{2}}+4V_{\omega_{1}}+5V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -1, 0, 0)(0, 0, 0, -1, 0, 0)g_{-4}-\varepsilon_{2}+\varepsilon_{3}
Module 22(0, 0, 0, -1, -1, 0)(0, 0, 1, 0, 0, 0)g_{3}
g_{-10}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{4}
Module 31(0, 0, 0, 1, 0, 0)(0, 0, 0, 1, 0, 0)g_{4}\varepsilon_{2}-\varepsilon_{3}
Module 42(0, 0, -1, -1, 0, 0)(0, 0, 0, 0, 1, 0)g_{5}
g_{-9}		\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{3}
Module 52(0, 0, 0, -1, -1, -1)(1, 0, 1, 0, 0, 0)g_{7}
g_{-16}		1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{5}
Module 62(0, 0, 0, 0, -1, 0)(0, 0, 1, 1, 0, 0)g_{9}
g_{-5}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{4}
Module 72(0, 0, -1, 0, 0, 0)(0, 0, 0, 1, 1, 0)g_{10}
g_{-3}		\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{2}
Module 82(-1, 0, -1, -1, 0, 0)(0, 0, 0, 0, 1, 1)g_{11}
g_{-12}		\varepsilon_{3}-\varepsilon_{5}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 92(0, 0, 0, 0, -1, -1)(1, 0, 1, 1, 0, 0)g_{12}
g_{-11}		1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{5}
Module 103(0, 0, -1, -1, -1, 0)(0, 0, 1, 1, 1, 0)g_{15}
h_{5}+h_{4}+h_{3}
g_{-15}		\varepsilon_{1}-\varepsilon_{4}
0
-\varepsilon_{1}+\varepsilon_{4}
Module 112(-1, 0, -1, 0, 0, 0)(0, 0, 0, 1, 1, 1)g_{16}
g_{-7}		\varepsilon_{2}-\varepsilon_{5}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 124(0, 0, -1, -1, -1, -1)(1, 0, 1, 1, 1, 0)g_{18}
g_{-6}
g_{1}
g_{-21}		1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{5}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{5}
Module 134(-1, 0, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1)g_{21}
g_{-1}
g_{6}
g_{-18}		\varepsilon_{1}-\varepsilon_{5}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
\varepsilon_{4}-\varepsilon_{5}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 142(0, -1, -1, -2, -1, -1)(1, 1, 1, 1, 1, 0)g_{22}
g_{-28}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{3}+\varepsilon_{5}
Module 153(-1, 0, -1, -1, -1, -1)(1, 0, 1, 1, 1, 1)g_{24}
h_{6}+h_{5}+h_{4}+h_{3}+h_{1}
g_{-24}		1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
0
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 162(-1, -1, -1, -2, -1, 0)(0, 1, 1, 1, 1, 1)g_{25}
g_{-26}		-\varepsilon_{2}-\varepsilon_{5}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 172(0, -1, -1, -1, -1, -1)(1, 1, 1, 2, 1, 0)g_{26}
g_{-25}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{5}
Module 182(-1, -1, -1, -1, -1, 0)(0, 1, 1, 2, 1, 1)g_{28}
g_{-22}		-\varepsilon_{3}-\varepsilon_{5}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 194(0, -1, -1, -2, -2, -1)(1, 1, 2, 2, 1, 0)g_{29}
g_{-20}
g_{17}
g_{-31}		1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{5}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{4}+\varepsilon_{5}
Module 204(-1, -1, -2, -2, -1, 0)(0, 1, 1, 2, 2, 1)g_{31}
g_{-17}
g_{20}
g_{-29}		-\varepsilon_{4}-\varepsilon_{5}
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{5}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 214(-1, -1, -1, -2, -2, -1)(1, 1, 2, 2, 1, 1)g_{32}
g_{-14}
g_{13}
g_{-33}		1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{3}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 224(-1, -1, -2, -2, -1, -1)(1, 1, 1, 2, 2, 1)g_{33}
g_{-13}
g_{14}
g_{-32}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 238(-1, -1, -2, -3, -2, -1)(1, 1, 2, 2, 2, 1)g_{34}
g_{-8}
g_{19}
g_{27}
g_{-30}
g_{-23}
g_{2}
g_{-35}		1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{4}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{2}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 248(-1, -1, -2, -2, -2, -1)(1, 1, 2, 3, 2, 1)g_{35}
g_{-2}
g_{23}
g_{30}
g_{-27}
g_{-19}
g_{8}
g_{-34}		1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{2}
-\varepsilon_{3}-\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{3}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 253(-1, -2, -2, -3, -2, -1)(1, 2, 2, 3, 2, 1)g_{36}
h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-36}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
0
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 261(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{4}0
Module 271(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}-h_{3}0
Module 281(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{6}-h_{1}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 23
Heirs rejected due to not being maximally dominant: 1
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 1
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,