Type: \(\displaystyle 5A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 5A^{1}_1\))
Simple basis: 5 vectors: (2, 2, 2, 2, 2, 2, 2, 1), (0, 2, 2, 2, 2, 2, 2, 1), (0, 0, 2, 2, 2, 2, 2, 1), (0, 0, 0, 2, 2, 2, 2, 1), (0, 0, 0, 0, 2, 2, 2, 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}: C^{1}_3
simple basis centralizer: 3 vectors: (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, 0, 1, 0)
Number of k-submodules of g: 66
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{5}}+V_{\omega_{4}+\omega_{5}}+V_{\omega_{3}+\omega_{5}}+V_{\omega_{2}+\omega_{5}}+V_{\omega_{1}+\omega_{5}}+V_{2\omega_{4}}+V_{\omega_{3}+\omega_{4}}+V_{\omega_{2}+\omega_{4}}+V_{\omega_{1}+\omega_{4}}+V_{2\omega_{3}}+V_{\omega_{2}+\omega_{3}}+V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+6V_{\omega_{5}}+6V_{\omega_{4}}+6V_{\omega_{3}}+6V_{\omega_{2}}+6V_{\omega_{1}}+21V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, 0, -2, -2, -1)(0, 0, 0, 0, 0, -2, -2, -1)g_{-34}-2\varepsilon_{6}
Module 21(0, 0, 0, 0, 0, -1, -2, -1)(0, 0, 0, 0, 0, -1, -2, -1)g_{-28}-\varepsilon_{6}-\varepsilon_{7}
Module 31(0, 0, 0, 0, 0, 0, -2, -1)(0, 0, 0, 0, 0, 0, -2, -1)g_{-22}-2\varepsilon_{7}
Module 41(0, 0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, 0, -1, -1, -1)g_{-21}-\varepsilon_{6}-\varepsilon_{8}
Module 51(0, 0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, 0, -1, -1)g_{-15}-\varepsilon_{7}-\varepsilon_{8}
Module 61(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, -1, -1, 0)g_{-14}-\varepsilon_{6}+\varepsilon_{8}
Module 71(0, 0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, 0, -1)g_{-8}-2\varepsilon_{8}
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, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 102(0, 0, 0, 0, -1, -2, -2, -1)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}
g_{-39}		\varepsilon_{5}-\varepsilon_{6}
-\varepsilon_{5}-\varepsilon_{6}
Module 111(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 121(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 131(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}2\varepsilon_{8}
Module 142(0, 0, 0, -1, -1, -2, -2, -1)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}
g_{-43}		\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
Module 152(0, 0, 0, 0, -1, -1, -2, -1)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}
g_{-33}		\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{5}-\varepsilon_{7}
Module 161(0, 0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}\varepsilon_{6}-\varepsilon_{8}
Module 171(0, 0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 0, 1, 1)g_{15}\varepsilon_{7}+\varepsilon_{8}
Module 182(0, 0, -1, -1, -1, -2, -2, -1)(0, 0, 1, 1, 1, 0, 0, 0)g_{18}
g_{-47}		\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 192(0, 0, 0, -1, -1, -1, -2, -1)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}
g_{-38}		\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
Module 202(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}
g_{-27}		\varepsilon_{5}-\varepsilon_{8}
-\varepsilon_{5}-\varepsilon_{8}
Module 211(0, 0, 0, 0, 0, 1, 1, 1)(0, 0, 0, 0, 0, 1, 1, 1)g_{21}\varepsilon_{6}+\varepsilon_{8}
Module 221(0, 0, 0, 0, 0, 0, 2, 1)(0, 0, 0, 0, 0, 0, 2, 1)g_{22}2\varepsilon_{7}
Module 232(0, -1, -1, -1, -1, -2, -2, -1)(0, 1, 1, 1, 1, 0, 0, 0)g_{24}
g_{-50}		\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
Module 242(0, 0, -1, -1, -1, -1, -2, -1)(0, 0, 1, 1, 1, 1, 0, 0)g_{25}
g_{-42}		\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 252(0, 0, 0, -1, -1, -1, -1, -1)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}
g_{-32}		\varepsilon_{4}-\varepsilon_{8}
-\varepsilon_{4}-\varepsilon_{8}
Module 262(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, 1, 1, 1, 1)g_{27}
g_{-20}		\varepsilon_{5}+\varepsilon_{8}
-\varepsilon_{5}+\varepsilon_{8}
Module 271(0, 0, 0, 0, 0, 1, 2, 1)(0, 0, 0, 0, 0, 1, 2, 1)g_{28}\varepsilon_{6}+\varepsilon_{7}
Module 282(-1, -1, -1, -1, -1, -2, -2, -1)(1, 1, 1, 1, 1, 0, 0, 0)g_{29}
g_{-53}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 292(0, -1, -1, -1, -1, -1, -2, -1)(0, 1, 1, 1, 1, 1, 0, 0)g_{30}
g_{-46}		\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
Module 302(0, 0, -1, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}
g_{-37}		\varepsilon_{3}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
Module 312(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 0, 1, 1, 1, 1, 1)g_{32}
g_{-26}		\varepsilon_{4}+\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{8}
Module 322(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, 1, 1, 2, 1)g_{33}
g_{-13}		\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{7}
Module 331(0, 0, 0, 0, 0, 2, 2, 1)(0, 0, 0, 0, 0, 2, 2, 1)g_{34}2\varepsilon_{6}
Module 342(-1, -1, -1, -1, -1, -1, -2, -1)(1, 1, 1, 1, 1, 1, 0, 0)g_{35}
g_{-49}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 352(0, -1, -1, -1, -1, -1, -1, -1)(0, 1, 1, 1, 1, 1, 1, 0)g_{36}
g_{-41}		\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
Module 362(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 1, 1)g_{37}
g_{-31}		\varepsilon_{3}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
Module 372(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 0, 1, 1, 1, 2, 1)g_{38}
g_{-19}		\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
Module 382(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, 1, 2, 2, 1)g_{39}
g_{-5}		\varepsilon_{5}+\varepsilon_{6}
-\varepsilon_{5}+\varepsilon_{6}
Module 392(-1, -1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-45}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
Module 402(0, -1, -1, -1, -1, -1, -1, 0)(0, 1, 1, 1, 1, 1, 1, 1)g_{41}
g_{-36}		\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
Module 412(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 1, 2, 1)g_{42}
g_{-25}		\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 422(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 0, 1, 1, 2, 2, 1)g_{43}
g_{-12}		\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
Module 433(0, 0, 0, 0, -2, -2, -2, -1)(0, 0, 0, 0, 2, 2, 2, 1)g_{44}
h_{8}+2h_{7}+2h_{6}+2h_{5}
g_{-44}		2\varepsilon_{5}
0
-2\varepsilon_{5}
Module 442(-1, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 1)g_{45}
g_{-40}		\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 452(0, -1, -1, -1, -1, -1, 0, 0)(0, 1, 1, 1, 1, 1, 2, 1)g_{46}
g_{-30}		\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
Module 462(0, 0, -1, -1, -1, 0, 0, 0)(0, 0, 1, 1, 1, 2, 2, 1)g_{47}
g_{-18}		\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
Module 474(0, 0, 0, -1, -2, -2, -2, -1)(0, 0, 0, 1, 2, 2, 2, 1)g_{48}
g_{-4}
g_{4}
g_{-48}		\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 482(-1, -1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 1, 2, 1)g_{49}
g_{-35}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 492(0, -1, -1, -1, -1, 0, 0, 0)(0, 1, 1, 1, 1, 2, 2, 1)g_{50}
g_{-24}		\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
Module 504(0, 0, -1, -1, -2, -2, -2, -1)(0, 0, 1, 1, 2, 2, 2, 1)g_{51}
g_{-11}
g_{11}
g_{-51}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 513(0, 0, 0, -2, -2, -2, -2, -1)(0, 0, 0, 2, 2, 2, 2, 1)g_{52}
h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}
g_{-52}		2\varepsilon_{4}
0
-2\varepsilon_{4}
Module 522(-1, -1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 1, 2, 2, 1)g_{53}
g_{-29}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 534(0, -1, -1, -1, -2, -2, -2, -1)(0, 1, 1, 1, 2, 2, 2, 1)g_{54}
g_{-17}
g_{17}
g_{-54}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 544(0, 0, -1, -2, -2, -2, -2, -1)(0, 0, 1, 2, 2, 2, 2, 1)g_{55}
g_{-3}
g_{3}
g_{-55}		\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 554(-1, -1, -1, -1, -2, -2, -2, -1)(1, 1, 1, 1, 2, 2, 2, 1)g_{56}
g_{-23}
g_{23}
g_{-56}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 564(0, -1, -1, -2, -2, -2, -2, -1)(0, 1, 1, 2, 2, 2, 2, 1)g_{57}
g_{-10}
g_{10}
g_{-57}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 573(0, 0, -2, -2, -2, -2, -2, -1)(0, 0, 2, 2, 2, 2, 2, 1)g_{58}
h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}
g_{-58}		2\varepsilon_{3}
0
-2\varepsilon_{3}
Module 584(-1, -1, -1, -2, -2, -2, -2, -1)(1, 1, 1, 2, 2, 2, 2, 1)g_{59}
g_{-16}
g_{16}
g_{-59}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 594(0, -1, -2, -2, -2, -2, -2, -1)(0, 1, 2, 2, 2, 2, 2, 1)g_{60}
g_{-2}
g_{2}
g_{-60}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 604(-1, -1, -2, -2, -2, -2, -2, -1)(1, 1, 2, 2, 2, 2, 2, 1)g_{61}
g_{-9}
g_{9}
g_{-61}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 613(0, -2, -2, -2, -2, -2, -2, -1)(0, 2, 2, 2, 2, 2, 2, 1)g_{62}
h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}
g_{-62}		2\varepsilon_{2}
0
-2\varepsilon_{2}
Module 624(-1, -2, -2, -2, -2, -2, -2, -1)(1, 2, 2, 2, 2, 2, 2, 1)g_{63}
g_{-1}
g_{1}
g_{-63}		\varepsilon_{1}+\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 633(-2, -2, -2, -2, -2, -2, -2, -1)(2, 2, 2, 2, 2, 2, 2, 1)g_{64}
h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}
g_{-64}		2\varepsilon_{1}
0
-2\varepsilon_{1}
Module 641(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 651(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 661(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: 57
Heirs rejected due to not being maximally dominant: 5
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 5
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by 4A^{1}_1
Potential Dynkin type extensions: 6A^{1}_1,