Type: \(\displaystyle 3A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 3A^{1}_1\))
Simple basis: 3 vectors: (2, 2, 2, 2, 2, 2, 1), (0, 2, 2, 2, 2, 2, 1), (0, 0, 2, 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}_4
simple basis centralizer: 4 vectors: (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of k-submodules of g: 66
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{3}}+V_{\omega_{2}+\omega_{3}}+V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+8V_{\omega_{3}}+8V_{\omega_{2}}+8V_{\omega_{1}}+36V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -2, -2, -2, -1)(0, 0, 0, -2, -2, -2, -1)g_{-37}-2\varepsilon_{4}
Module 21(0, 0, 0, -1, -2, -2, -1)(0, 0, 0, -1, -2, -2, -1)g_{-33}-\varepsilon_{4}-\varepsilon_{5}
Module 31(0, 0, 0, 0, -2, -2, -1)(0, 0, 0, 0, -2, -2, -1)g_{-29}-2\varepsilon_{5}
Module 41(0, 0, 0, -1, -1, -2, -1)(0, 0, 0, -1, -1, -2, -1)g_{-28}-\varepsilon_{4}-\varepsilon_{6}
Module 51(0, 0, 0, 0, -1, -2, -1)(0, 0, 0, 0, -1, -2, -1)g_{-24}-\varepsilon_{5}-\varepsilon_{6}
Module 61(0, 0, 0, -1, -1, -1, -1)(0, 0, 0, -1, -1, -1, -1)g_{-23}-\varepsilon_{4}-\varepsilon_{7}
Module 71(0, 0, 0, 0, 0, -2, -1)(0, 0, 0, 0, 0, -2, -1)g_{-19}-2\varepsilon_{6}
Module 81(0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, -1, -1, -1)g_{-18}-\varepsilon_{5}-\varepsilon_{7}
Module 91(0, 0, 0, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, 0)g_{-17}-\varepsilon_{4}+\varepsilon_{7}
Module 101(0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, -1, -1)g_{-13}-\varepsilon_{6}-\varepsilon_{7}
Module 111(0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, -1, -1, 0)g_{-12}-\varepsilon_{5}+\varepsilon_{7}
Module 121(0, 0, 0, -1, -1, 0, 0)(0, 0, 0, -1, -1, 0, 0)g_{-11}-\varepsilon_{4}+\varepsilon_{6}
Module 131(0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, -1)g_{-7}-2\varepsilon_{7}
Module 141(0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, -1, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 151(0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, -1, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 161(0, 0, 0, -1, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 172(0, 0, -1, -2, -2, -2, -1)(0, 0, 1, 0, 0, 0, 0)g_{3}
g_{-40}		\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 181(0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 191(0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 1, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 201(0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 1, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 211(0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 1)g_{7}2\varepsilon_{7}
Module 222(0, -1, -1, -2, -2, -2, -1)(0, 1, 1, 0, 0, 0, 0)g_{9}
g_{-42}		\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 232(0, 0, -1, -1, -2, -2, -1)(0, 0, 1, 1, 0, 0, 0)g_{10}
g_{-36}		\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 241(0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 1, 1, 0, 0)g_{11}\varepsilon_{4}-\varepsilon_{6}
Module 251(0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 1, 1, 0)g_{12}\varepsilon_{5}-\varepsilon_{7}
Module 261(0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 1, 1)g_{13}\varepsilon_{6}+\varepsilon_{7}
Module 272(-1, -1, -1, -2, -2, -2, -1)(1, 1, 1, 0, 0, 0, 0)g_{14}
g_{-44}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 282(0, -1, -1, -1, -2, -2, -1)(0, 1, 1, 1, 0, 0, 0)g_{15}
g_{-39}		\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 292(0, 0, -1, -1, -1, -2, -1)(0, 0, 1, 1, 1, 0, 0)g_{16}
g_{-32}		\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 301(0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 0)g_{17}\varepsilon_{4}-\varepsilon_{7}
Module 311(0, 0, 0, 0, 1, 1, 1)(0, 0, 0, 0, 1, 1, 1)g_{18}\varepsilon_{5}+\varepsilon_{7}
Module 321(0, 0, 0, 0, 0, 2, 1)(0, 0, 0, 0, 0, 2, 1)g_{19}2\varepsilon_{6}
Module 332(-1, -1, -1, -1, -2, -2, -1)(1, 1, 1, 1, 0, 0, 0)g_{20}
g_{-41}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 342(0, -1, -1, -1, -1, -2, -1)(0, 1, 1, 1, 1, 0, 0)g_{21}
g_{-35}		\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
Module 352(0, 0, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 0)g_{22}
g_{-27}		\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 361(0, 0, 0, 1, 1, 1, 1)(0, 0, 0, 1, 1, 1, 1)g_{23}\varepsilon_{4}+\varepsilon_{7}
Module 371(0, 0, 0, 0, 1, 2, 1)(0, 0, 0, 0, 1, 2, 1)g_{24}\varepsilon_{5}+\varepsilon_{6}
Module 382(-1, -1, -1, -1, -1, -2, -1)(1, 1, 1, 1, 1, 0, 0)g_{25}
g_{-38}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 392(0, -1, -1, -1, -1, -1, -1)(0, 1, 1, 1, 1, 1, 0)g_{26}
g_{-31}		\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
Module 402(0, 0, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 1)g_{27}
g_{-22}		\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 411(0, 0, 0, 1, 1, 2, 1)(0, 0, 0, 1, 1, 2, 1)g_{28}\varepsilon_{4}+\varepsilon_{6}
Module 421(0, 0, 0, 0, 2, 2, 1)(0, 0, 0, 0, 2, 2, 1)g_{29}2\varepsilon_{5}
Module 432(-1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 0)g_{30}
g_{-34}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 442(0, -1, -1, -1, -1, -1, 0)(0, 1, 1, 1, 1, 1, 1)g_{31}
g_{-26}		\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
Module 452(0, 0, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 2, 1)g_{32}
g_{-16}		\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
Module 461(0, 0, 0, 1, 2, 2, 1)(0, 0, 0, 1, 2, 2, 1)g_{33}\varepsilon_{4}+\varepsilon_{5}
Module 472(-1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1)g_{34}
g_{-30}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 482(0, -1, -1, -1, -1, 0, 0)(0, 1, 1, 1, 1, 2, 1)g_{35}
g_{-21}		\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
Module 492(0, 0, -1, -1, 0, 0, 0)(0, 0, 1, 1, 2, 2, 1)g_{36}
g_{-10}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
Module 501(0, 0, 0, 2, 2, 2, 1)(0, 0, 0, 2, 2, 2, 1)g_{37}2\varepsilon_{4}
Module 512(-1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 2, 1)g_{38}
g_{-25}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 522(0, -1, -1, -1, 0, 0, 0)(0, 1, 1, 1, 2, 2, 1)g_{39}
g_{-15}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
Module 532(0, 0, -1, 0, 0, 0, 0)(0, 0, 1, 2, 2, 2, 1)g_{40}
g_{-3}		\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
Module 542(-1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 2, 2, 1)g_{41}
g_{-20}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 552(0, -1, -1, 0, 0, 0, 0)(0, 1, 1, 2, 2, 2, 1)g_{42}
g_{-9}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
Module 563(0, 0, -2, -2, -2, -2, -1)(0, 0, 2, 2, 2, 2, 1)g_{43}
h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}
g_{-43}		2\varepsilon_{3}
0
-2\varepsilon_{3}
Module 572(-1, -1, -1, 0, 0, 0, 0)(1, 1, 1, 2, 2, 2, 1)g_{44}
g_{-14}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 584(0, -1, -2, -2, -2, -2, -1)(0, 1, 2, 2, 2, 2, 1)g_{45}
g_{-2}
g_{2}
g_{-45}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 594(-1, -1, -2, -2, -2, -2, -1)(1, 1, 2, 2, 2, 2, 1)g_{46}
g_{-8}
g_{8}
g_{-46}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 603(0, -2, -2, -2, -2, -2, -1)(0, 2, 2, 2, 2, 2, 1)g_{47}
h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}
g_{-47}		2\varepsilon_{2}
0
-2\varepsilon_{2}
Module 614(-1, -2, -2, -2, -2, -2, -1)(1, 2, 2, 2, 2, 2, 1)g_{48}
g_{-1}
g_{1}
g_{-48}		\varepsilon_{1}+\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 623(-2, -2, -2, -2, -2, -2, -1)(2, 2, 2, 2, 2, 2, 1)g_{49}
h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}
g_{-49}		2\varepsilon_{1}
0
-2\varepsilon_{1}
Module 631(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 641(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 651(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 661(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: 54
Heirs rejected due to not being maximally dominant: 7
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 7
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,