Type: \(\displaystyle 6A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 6A^{1}_1\))
Simple basis: 6 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), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 2, 2, 2), (0, 0, 0, 0, 1, 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}_2
simple basis centralizer: 2 vectors: (0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, 0, 1, 0)
Number of k-submodules of g: 34
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{3}+\omega_{4}+\omega_{5}+\omega_{6}}+V_{\omega_{1}+\omega_{2}+\omega_{5}+\omega_{6}}+V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}+V_{2\omega_{6}}+5V_{\omega_{5}+\omega_{6}}+V_{2\omega_{5}}+V_{2\omega_{4}}+5V_{\omega_{3}+\omega_{4}}+V_{2\omega_{3}}+V_{2\omega_{2}}+5V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+10V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, 0, -1, -2)g_{-22}-\varepsilon_{7}-\varepsilon_{8}
Module 21(0, 0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, 0, -1, -1)g_{-15}-\varepsilon_{7}
Module 31(0, 0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, 0, -1)g_{-8}-\varepsilon_{8}
Module 41(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 53(-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 63(0, 0, -1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}
h_{3}
g_{-3}		\varepsilon_{3}-\varepsilon_{4}
0
-\varepsilon_{3}+\varepsilon_{4}
Module 73(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}
h_{5}
g_{-5}		\varepsilon_{5}-\varepsilon_{6}
0
-\varepsilon_{5}+\varepsilon_{6}
Module 81(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 91(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}\varepsilon_{8}
Module 104(0, 0, 0, 0, -1, -1, -2, -2)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}
g_{-34}
g_{6}
g_{-39}		\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{6}-\varepsilon_{7}
\varepsilon_{6}-\varepsilon_{7}
-\varepsilon_{5}-\varepsilon_{7}
Module 111(0, 0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 0, 1, 1)g_{15}\varepsilon_{7}
Module 124(0, 0, 0, 0, -1, -1, -1, -2)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}
g_{-28}
g_{14}
g_{-33}		\varepsilon_{5}-\varepsilon_{8}
-\varepsilon_{6}-\varepsilon_{8}
\varepsilon_{6}-\varepsilon_{8}
-\varepsilon_{5}-\varepsilon_{8}
Module 131(0, 0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 0, 1, 2)g_{22}\varepsilon_{7}+\varepsilon_{8}
Module 144(0, 0, -1, -1, -1, -1, -2, -2)(0, 0, 1, 1, 1, 1, 0, 0)g_{25}
g_{-43}
g_{19}
g_{-47}		\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 154(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, 1, 1, 1, 1)g_{27}
g_{-21}
g_{21}
g_{-27}		\varepsilon_{5}
-\varepsilon_{6}
\varepsilon_{6}
-\varepsilon_{5}
Module 164(0, 0, -1, -1, -1, -1, -1, -2)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}
g_{-38}
g_{26}
g_{-42}		\varepsilon_{3}-\varepsilon_{8}
-\varepsilon_{4}-\varepsilon_{8}
\varepsilon_{4}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
Module 174(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, 1, 1, 1, 2)g_{33}
g_{-14}
g_{28}
g_{-20}		\varepsilon_{5}+\varepsilon_{8}
-\varepsilon_{6}+\varepsilon_{8}
\varepsilon_{6}+\varepsilon_{8}
-\varepsilon_{5}+\varepsilon_{8}
Module 184(-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 194(0, 0, -1, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 1, 1)g_{37}
g_{-32}
g_{32}
g_{-37}		\varepsilon_{3}
-\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{3}
Module 204(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, 1, 1, 2, 2)g_{39}
g_{-6}
g_{34}
g_{-13}		\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{6}+\varepsilon_{7}
\varepsilon_{6}+\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{7}
Module 214(-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 224(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 1, 2)g_{42}
g_{-26}
g_{38}
g_{-31}		\varepsilon_{3}+\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{8}
\varepsilon_{4}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
Module 233(0, 0, 0, 0, -1, -2, -2, -2)(0, 0, 0, 0, 1, 2, 2, 2)g_{44}
2h_{8}+2h_{7}+2h_{6}+h_{5}
g_{-44}		\varepsilon_{5}+\varepsilon_{6}
0
-\varepsilon_{5}-\varepsilon_{6}
Module 244(-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 254(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 1, 2, 2)g_{47}
g_{-19}
g_{43}
g_{-25}		\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 264(-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 274(-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 2816(0, 0, -1, -1, -2, -2, -2, -2)(0, 0, 1, 1, 2, 2, 2, 2)g_{55}
g_{-4}
g_{52}
g_{18}
g_{51}
g_{-11}
g_{-48}
g_{12}
g_{-12}
g_{48}
g_{11}
g_{-51}
g_{-18}
g_{-52}
g_{4}
g_{-55}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 293(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 3016(-1, -1, -1, -1, -2, -2, -2, -2)(1, 1, 1, 1, 2, 2, 2, 2)g_{59}
g_{-17}
g_{57}
g_{29}
g_{56}
g_{-23}
g_{-54}
g_{24}
g_{-24}
g_{54}
g_{23}
g_{-56}
g_{-29}
g_{-57}
g_{17}
g_{-59}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{6}
\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 3116(-1, -1, -2, -2, -2, -2, -2, -2)(1, 1, 2, 2, 2, 2, 2, 2)g_{63}
g_{-2}
g_{62}
g_{16}
g_{61}
g_{-9}
g_{-60}
g_{10}
g_{-10}
g_{60}
g_{9}
g_{-61}
g_{-16}
g_{-62}
g_{2}
g_{-63}		\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 323(-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 331(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 341(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: 28
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: 0
Parabolically induced by 5A^{1}_1
Potential Dynkin type extensions: 7A^{1}_1,