Type: \(\displaystyle 4A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 4A^{1}_1\))
Simple basis: 4 vectors: (1, 2, 2, 2, 2, 2, 2), (1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 2, 2, 2, 2), (0, 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}: B^{1}_3
simple basis centralizer: 3 vectors: (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1)
Number of k-submodules of g: 40
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}+V_{2\omega_{4}}+7V_{\omega_{3}+\omega_{4}}+V_{2\omega_{3}}+V_{2\omega_{2}}+7V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+21V_{0}\)
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)(0, 0, 0, 0, -1, -2, -2)g_{-29}-\varepsilon_{5}-\varepsilon_{6}
Module 21(0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, -1, -1, -2)g_{-24}-\varepsilon_{5}-\varepsilon_{7}
Module 31(0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, -1, -2)g_{-19}-\varepsilon_{6}-\varepsilon_{7}
Module 41(0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, -1, -1, -1)g_{-18}-\varepsilon_{5}
Module 51(0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, -1, -1)g_{-13}-\varepsilon_{6}
Module 61(0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, -1, -1, 0)g_{-12}-\varepsilon_{5}+\varepsilon_{7}
Module 71(0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, -1)g_{-7}-\varepsilon_{7}
Module 81(0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, -1, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 91(0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, -1, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 103(-1, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0)g_{1}
h_{1}
g_{-1}		\varepsilon_{1}-\varepsilon_{2}
0
-\varepsilon_{1}+\varepsilon_{2}
Module 113(0, 0, -1, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0)g_{3}
h_{3}
g_{-3}		\varepsilon_{3}-\varepsilon_{4}
0
-\varepsilon_{3}+\varepsilon_{4}
Module 121(0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 1, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 131(0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 1, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 141(0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 1)g_{7}\varepsilon_{7}
Module 154(0, 0, -1, -1, -2, -2, -2)(0, 0, 1, 1, 0, 0, 0)g_{10}
g_{-37}
g_{4}
g_{-40}		\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 161(0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 1, 1, 0)g_{12}\varepsilon_{5}-\varepsilon_{7}
Module 171(0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 1, 1)g_{13}\varepsilon_{6}
Module 184(0, 0, -1, -1, -1, -2, -2)(0, 0, 1, 1, 1, 0, 0)g_{16}
g_{-33}
g_{11}
g_{-36}		\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 191(0, 0, 0, 0, 1, 1, 1)(0, 0, 0, 0, 1, 1, 1)g_{18}\varepsilon_{5}
Module 201(0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 1, 2)g_{19}\varepsilon_{6}+\varepsilon_{7}
Module 214(-1, -1, -1, -1, -2, -2, -2)(1, 1, 1, 1, 0, 0, 0)g_{20}
g_{-42}
g_{15}
g_{-44}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 224(0, 0, -1, -1, -1, -1, -2)(0, 0, 1, 1, 1, 1, 0)g_{22}
g_{-28}
g_{17}
g_{-32}		\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 231(0, 0, 0, 0, 1, 1, 2)(0, 0, 0, 0, 1, 1, 2)g_{24}\varepsilon_{5}+\varepsilon_{7}
Module 244(-1, -1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 1, 0, 0)g_{25}
g_{-39}
g_{21}
g_{-41}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 254(0, 0, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 1)g_{27}
g_{-23}
g_{23}
g_{-27}		\varepsilon_{3}
-\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{3}
Module 261(0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 1, 2, 2)g_{29}\varepsilon_{5}+\varepsilon_{6}
Module 274(-1, -1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 1, 0)g_{30}
g_{-35}
g_{26}
g_{-38}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 284(0, 0, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 2)g_{32}
g_{-17}
g_{28}
g_{-22}		\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 294(-1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1)g_{34}
g_{-31}
g_{31}
g_{-34}		\varepsilon_{1}
-\varepsilon_{2}
\varepsilon_{2}
-\varepsilon_{1}
Module 304(0, 0, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 2, 2)g_{36}
g_{-11}
g_{33}
g_{-16}		\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
Module 314(-1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 2)g_{38}
g_{-26}
g_{35}
g_{-30}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 324(0, 0, -1, -1, 0, 0, 0)(0, 0, 1, 1, 2, 2, 2)g_{40}
g_{-4}
g_{37}
g_{-10}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
Module 334(-1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 2, 2)g_{41}
g_{-21}
g_{39}
g_{-25}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 343(0, 0, -1, -2, -2, -2, -2)(0, 0, 1, 2, 2, 2, 2)g_{43}
2h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}
g_{-43}		\varepsilon_{3}+\varepsilon_{4}
0
-\varepsilon_{3}-\varepsilon_{4}
Module 354(-1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 2, 2, 2)g_{44}
g_{-15}
g_{42}
g_{-20}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 3616(-1, -1, -2, -2, -2, -2, -2)(1, 1, 2, 2, 2, 2, 2)g_{48}
g_{-2}
g_{47}
g_{14}
g_{46}
g_{-8}
g_{-45}
g_{9}
g_{-9}
g_{45}
g_{8}
g_{-46}
g_{-14}
g_{-47}
g_{2}
g_{-48}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 373(-1, -2, -2, -2, -2, -2, -2)(1, 2, 2, 2, 2, 2, 2)g_{49}
2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-49}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 381(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 391(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 401(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{7}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 25
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 3A^{1}_1
Potential Dynkin type extensions: 5A^{1}_1,