Type: \(\displaystyle 3A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 3A^{1}_1\))
Simple basis: 3 vectors: (1, 2, 2, 2, 2, 2, 1, 1), (0, 0, 1, 2, 2, 2, 1, 1), (0, 0, 0, 0, 1, 2, 1, 1)
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}: 5A^{1}_1
simple basis centralizer: 5 vectors: (0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0)
Number of k-submodules of g: 54
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{3}}+4V_{\omega_{2}+\omega_{3}}+4V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+4V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+8V_{\omega_{3}}+8V_{\omega_{2}}+8V_{\omega_{1}}+15V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, 0, -1)g_{-8}-\varepsilon_{7}-\varepsilon_{8}
Module 21(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 31(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 41(0, 0, -1, 0, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 51(-1, 0, 0, 0, 0, 0, 0, 0)(-1, 0, 0, 0, 0, 0, 0, 0)g_{-1}-\varepsilon_{1}+\varepsilon_{2}
Module 61(1, 0, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}\varepsilon_{1}-\varepsilon_{2}
Module 71(0, 0, 1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 81(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 92(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}
g_{-28}		\varepsilon_{6}-\varepsilon_{7}
-\varepsilon_{5}-\varepsilon_{7}
Module 101(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 111(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}\varepsilon_{7}+\varepsilon_{8}
Module 122(0, 0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}
g_{-22}		\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{6}-\varepsilon_{7}
Module 132(0, 0, 0, 0, -1, -1, 0, -1)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}
g_{-21}		\varepsilon_{6}-\varepsilon_{8}
-\varepsilon_{5}-\varepsilon_{8}
Module 142(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, 0, 1, 0, 1)g_{15}
g_{-20}		\varepsilon_{6}+\varepsilon_{8}
-\varepsilon_{5}+\varepsilon_{8}
Module 152(0, 0, -1, -1, -1, -1, -1, -1)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}
g_{-38}		\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 162(0, 0, 0, 0, 0, -1, 0, -1)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}
g_{-15}		\varepsilon_{5}-\varepsilon_{8}
-\varepsilon_{6}-\varepsilon_{8}
Module 172(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 1, 1, 0, 1)g_{21}
g_{-14}		\varepsilon_{5}+\varepsilon_{8}
-\varepsilon_{6}+\varepsilon_{8}
Module 182(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, 0, 1, 1, 1)g_{22}
g_{-13}		\varepsilon_{6}+\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{7}
Module 192(0, 0, 0, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 0, 0)g_{25}
g_{-33}		\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
Module 202(0, 0, -1, -1, -1, -1, 0, -1)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}
g_{-32}		\varepsilon_{4}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
Module 212(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, 0, 1, 1, 1, 0, 1)g_{27}
g_{-31}		\varepsilon_{4}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
Module 222(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 1, 1, 1, 1)g_{28}
g_{-6}		\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{6}+\varepsilon_{7}
Module 232(-1, -1, -1, -1, -1, -1, -1, -1)(0, 1, 1, 1, 1, 1, 0, 0)g_{30}
g_{-45}		\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 242(0, 0, 0, -1, -1, -1, 0, -1)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}
g_{-27}		\varepsilon_{3}-\varepsilon_{8}
-\varepsilon_{4}-\varepsilon_{8}
Module 252(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 0, 1)g_{32}
g_{-26}		\varepsilon_{3}+\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{8}
Module 262(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, 0, 1, 1, 1, 1, 1)g_{33}
g_{-25}		\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 273(0, 0, 0, 0, -1, -2, -1, -1)(0, 0, 0, 0, 1, 2, 1, 1)g_{34}
h_{8}+h_{7}+2h_{6}+h_{5}
g_{-34}		\varepsilon_{5}+\varepsilon_{6}
0
-\varepsilon_{5}-\varepsilon_{6}
Module 282(0, -1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 0, 0)g_{35}
g_{-42}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
Module 292(-1, -1, -1, -1, -1, -1, 0, -1)(0, 1, 1, 1, 1, 1, 1, 0)g_{36}
g_{-41}		\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
Module 302(-1, -1, -1, -1, -1, -1, -1, 0)(0, 1, 1, 1, 1, 1, 0, 1)g_{37}
g_{-40}		\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 312(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 1, 1, 1)g_{38}
g_{-19}		\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
Module 324(0, 0, -1, -1, -2, -2, -1, -1)(0, 0, 0, 1, 1, 2, 1, 1)g_{39}
g_{-18}
g_{4}
g_{-47}		\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 332(0, -1, -1, -1, -1, -1, 0, -1)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-37}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
Module 342(0, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 0, 1)g_{41}
g_{-36}		\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
Module 352(-1, -1, -1, -1, -1, -1, 0, 0)(0, 1, 1, 1, 1, 1, 1, 1)g_{42}
g_{-35}		\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 364(0, 0, 0, -1, -2, -2, -1, -1)(0, 0, 1, 1, 1, 2, 1, 1)g_{43}
g_{-12}
g_{11}
g_{-44}		\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 374(0, 0, -1, -1, -1, -2, -1, -1)(0, 0, 0, 1, 2, 2, 1, 1)g_{44}
g_{-11}
g_{12}
g_{-43}		\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 382(0, -1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 1, 1, 1)g_{45}
g_{-30}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
Module 394(-1, -1, -1, -1, -2, -2, -1, -1)(0, 1, 1, 1, 1, 2, 1, 1)g_{46}
g_{-29}
g_{17}
g_{-51}		\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 404(0, 0, 0, -1, -1, -2, -1, -1)(0, 0, 1, 1, 2, 2, 1, 1)g_{47}
g_{-4}
g_{18}
g_{-39}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
Module 414(0, -1, -1, -1, -2, -2, -1, -1)(1, 1, 1, 1, 1, 2, 1, 1)g_{48}
g_{-24}
g_{23}
g_{-49}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 424(-1, -1, -1, -1, -1, -2, -1, -1)(0, 1, 1, 1, 2, 2, 1, 1)g_{49}
g_{-23}
g_{24}
g_{-48}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 433(0, 0, -1, -2, -2, -2, -1, -1)(0, 0, 1, 2, 2, 2, 1, 1)g_{50}
h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}
g_{-50}		\varepsilon_{3}+\varepsilon_{4}
0
-\varepsilon_{3}-\varepsilon_{4}
Module 444(0, -1, -1, -1, -1, -2, -1, -1)(1, 1, 1, 1, 2, 2, 1, 1)g_{51}
g_{-17}
g_{29}
g_{-46}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
Module 454(-1, -1, -2, -2, -2, -2, -1, -1)(0, 1, 1, 2, 2, 2, 1, 1)g_{52}
g_{-16}
g_{2}
g_{-55}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 464(0, -1, -2, -2, -2, -2, -1, -1)(1, 1, 1, 2, 2, 2, 1, 1)g_{53}
g_{-10}
g_{9}
g_{-54}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 474(-1, -1, -1, -2, -2, -2, -1, -1)(0, 1, 2, 2, 2, 2, 1, 1)g_{54}
g_{-9}
g_{10}
g_{-53}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 484(0, -1, -1, -2, -2, -2, -1, -1)(1, 1, 2, 2, 2, 2, 1, 1)g_{55}
g_{-2}
g_{16}
g_{-52}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 493(-1, -2, -2, -2, -2, -2, -1, -1)(1, 2, 2, 2, 2, 2, 1, 1)g_{56}
h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-56}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 501(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{1}0
Module 511(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 521(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 531(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 541(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: 39
Heirs rejected due to not being maximally dominant: 6
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 6
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,