Type: \(\displaystyle A^{1}_2+A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_2+A^{1}_1\))
Simple basis: 3 vectors: (1, 2, 2, 3, 2, 1), (0, -1, 0, 0, 0, 0), (1, 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}: A^{1}_2
simple basis centralizer: 2 vectors: (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1)
Number of k-submodules of g: 25
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{3}}+3V_{\omega_{2}+\omega_{3}}+3V_{\omega_{1}+\omega_{3}}+V_{\omega_{1}+\omega_{2}}+2V_{\omega_{3}}+3V_{\omega_{2}}+3V_{\omega_{1}}+9V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, -1, -1)(0, 0, 0, 0, -1, -1)g_{-11}-\varepsilon_{3}+\varepsilon_{5}
Module 21(0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, -1)g_{-6}-\varepsilon_{4}+\varepsilon_{5}
Module 31(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{3}+\varepsilon_{4}
Module 42(0, 0, -1, 0, 0, 0)(1, 0, 0, 0, 0, 0)g_{1}
g_{-3}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{2}
Module 52(-1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0)g_{3}
g_{-1}		\varepsilon_{1}-\varepsilon_{2}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 61(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{3}-\varepsilon_{4}
Module 71(0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 1)g_{6}\varepsilon_{4}-\varepsilon_{5}
Module 83(-1, 0, -1, 0, 0, 0)(1, 0, 1, 0, 0, 0)g_{7}
h_{3}+h_{1}
g_{-7}		1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
0
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 93(-1, -1, -1, -2, -2, -1)(0, 0, 1, 1, 0, 0)g_{9}
g_{13}
g_{-33}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 101(0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 1, 1)g_{11}\varepsilon_{3}-\varepsilon_{5}
Module 116(-1, -1, -2, -2, -2, -1)(1, 0, 1, 1, 0, 0)g_{12}
g_{17}
g_{4}
g_{-31}
g_{8}
g_{-34}		1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{3}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 123(-1, -1, -1, -2, -1, -1)(0, 0, 1, 1, 1, 0)g_{15}
g_{19}
g_{-30}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 136(-1, -1, -2, -2, -1, -1)(1, 0, 1, 1, 1, 0)g_{18}
g_{22}
g_{10}
g_{-28}
g_{14}
g_{-32}		1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 143(-1, -1, -1, -2, -1, 0)(0, 0, 1, 1, 1, 1)g_{21}
g_{25}
g_{-26}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 156(-1, -1, -2, -2, -1, 0)(1, 0, 1, 1, 1, 1)g_{24}
g_{27}
g_{16}
g_{-23}
g_{20}
g_{-29}		1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{5}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 163(0, 0, -1, -1, -1, -1)(1, 1, 1, 2, 1, 0)g_{26}
g_{-25}
g_{-21}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 176(-1, 0, -1, -1, -1, -1)(1, 1, 2, 2, 1, 0)g_{29}
g_{-20}
g_{23}
g_{-16}
g_{-27}
g_{-24}		1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{5}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 183(0, 0, -1, -1, -1, 0)(1, 1, 1, 2, 1, 1)g_{30}
g_{-19}
g_{-15}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 196(-1, 0, -1, -1, -1, 0)(1, 1, 2, 2, 1, 1)g_{32}
g_{-14}
g_{28}
g_{-10}
g_{-22}
g_{-18}		1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{4}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 203(0, 0, -1, -1, 0, 0)(1, 1, 1, 2, 2, 1)g_{33}
g_{-13}
g_{-9}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 216(-1, 0, -1, -1, 0, 0)(1, 1, 2, 2, 2, 1)g_{34}
g_{-8}
g_{31}
g_{-4}
g_{-17}
g_{-12}		1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{3}
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 228(-1, -1, -2, -3, -2, -1)(1, 1, 2, 3, 2, 1)g_{35}
g_{-2}
g_{36}
-h_{2}
h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-36}
g_{2}
g_{-35}		1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{2}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
0
0
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{2}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 231(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{3}-h_{1}0
Module 241(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Module 251(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{6}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 14
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: 1
Parabolically induced by A^{1}_2
Potential Dynkin type extensions: 2A^{1}_2, A^{1}_2+2A^{1}_1,