sp(12), type \(C^{1}_6\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: C^{1}_6.
Weyl group size: 46080.
A drawing of the root system in its corresponding Coxeter plane. Computations were carried out as explained by John Stembridge.
The darker red dots can be dragged with the mouse to rotate the picture.
The grey lines are the edges of the Weyl chamber.
Canvas not supported


The root system has 72 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-2, -2, -2, -2, -2, -1)-2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(-1, -2, -2, -2, -2, -1)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, -2, -2, -2, -2, -1)-2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -2, -1)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -2, -2, -2, -1)-e_{2}-e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(-1, -1, -1, -2, -2, -1)-e_{1}-e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}\)
(0, 0, -2, -2, -2, -1)-2e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -2, -2, -1)-e_{2}-e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}\)
(-1, -1, -1, -1, -2, -1)-e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}\)
(0, 0, -1, -2, -2, -1)-e_{3}-e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}\)
(0, -1, -1, -1, -2, -1)-e_{2}-e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}\)
(-1, -1, -1, -1, -1, -1)-e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}\)
(0, 0, 0, -2, -2, -1)-2e_{4}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -2, -1)-e_{3}-e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{6}s_{5}\)
(0, -1, -1, -1, -1, -1)-e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}\)
(-1, -1, -1, -1, -1, 0)-e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, -1, -2, -1)-e_{4}-e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{4}s_{6}s_{5}\)
(0, 0, -1, -1, -1, -1)-e_{3}-e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{4}s_{3}s_{6}\)
(0, -1, -1, -1, -1, 0)-e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, 0, 0)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -2, -1)-2e_{5}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1)-e_{4}-e_{6}\(s_{4}s_{6}s_{5}s_{4}s_{6}\)
(0, 0, -1, -1, -1, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -1)-e_{5}-e_{6}\(s_{6}s_{5}s_{6}\)
(0, 0, 0, -1, -1, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1)-2e_{6}\(s_{6}\)
(0, 0, 0, 0, -1, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1)2e_{6}\(s_{6}\)
(1, 1, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1)e_{5}+e_{6}\(s_{6}s_{5}s_{6}\)
(1, 1, 1, 0, 0, 0)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 0, 0)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1)e_{4}+e_{6}\(s_{4}s_{6}s_{5}s_{4}s_{6}\)
(0, 0, 0, 0, 2, 1)2e_{5}\(s_{5}s_{6}s_{5}\)
(1, 1, 1, 1, 0, 0)e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 0)e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1)e_{3}+e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{4}s_{3}s_{6}\)
(0, 0, 0, 1, 2, 1)e_{4}+e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{4}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 0)e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1)e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}\)
(0, 0, 1, 1, 2, 1)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{6}s_{5}\)
(0, 0, 0, 2, 2, 1)2e_{4}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 1, 1)e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}\)
(0, 1, 1, 1, 2, 1)e_{2}+e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}\)
(0, 0, 1, 2, 2, 1)e_{3}+e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 2, 1)e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}\)
(0, 1, 1, 2, 2, 1)e_{2}+e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}\)
(0, 0, 2, 2, 2, 1)2e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 1)e_{1}+e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}\)
(0, 1, 2, 2, 2, 1)e_{2}+e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 2, 1)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}\)
(0, 2, 2, 2, 2, 1)2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(1, 2, 2, 2, 2, 1)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(2, 2, 2, 2, 2, 1)2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-2, -2, -2, -2, -2, -1), (-1, -2, -2, -2, -2, -1), (0, -2, -2, -2, -2, -1), (-1, -1, -2, -2, -2, -1), (0, -1, -2, -2, -2, -1), (-1, -1, -1, -2, -2, -1), (0, 0, -2, -2, -2, -1), (0, -1, -1, -2, -2, -1), (-1, -1, -1, -1, -2, -1), (0, 0, -1, -2, -2, -1), (0, -1, -1, -1, -2, -1), (-1, -1, -1, -1, -1, -1), (0, 0, 0, -2, -2, -1), (0, 0, -1, -1, -2, -1), (0, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, 0), (0, 0, 0, -1, -2, -1), (0, 0, -1, -1, -1, -1), (0, -1, -1, -1, -1, 0), (-1, -1, -1, -1, 0, 0), (0, 0, 0, 0, -2, -1), (0, 0, 0, -1, -1, -1), (0, 0, -1, -1, -1, 0), (0, -1, -1, -1, 0, 0), (-1, -1, -1, 0, 0, 0), (0, 0, 0, 0, -1, -1), (0, 0, 0, -1, -1, 0), (0, 0, -1, -1, 0, 0), (0, -1, -1, 0, 0, 0), (-1, -1, 0, 0, 0, 0), (0, 0, 0, 0, 0, -1), (0, 0, 0, 0, -1, 0), (0, 0, 0, -1, 0, 0), (0, 0, -1, 0, 0, 0), (0, -1, 0, 0, 0, 0), (-1, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1), (1, 1, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0), (0, 0, 1, 1, 0, 0), (0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 1, 1), (1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1), (0, 0, 0, 0, 2, 1), (1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1), (0, 0, 0, 1, 2, 1), (1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1), (0, 0, 1, 1, 2, 1), (0, 0, 0, 2, 2, 1), (1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 2, 1), (0, 0, 1, 2, 2, 1), (1, 1, 1, 1, 2, 1), (0, 1, 1, 2, 2, 1), (0, 0, 2, 2, 2, 1), (1, 1, 1, 2, 2, 1), (0, 1, 2, 2, 2, 1), (1, 1, 2, 2, 2, 1), (0, 2, 2, 2, 2, 1), (1, 2, 2, 2, 2, 1), (2, 2, 2, 2, 2, 1) The resulting Lie bracket pairing table follows.
Type C^{1}_6.The letter \(\displaystyle h\) stands for elements of the Cartan subalgebra,
the letter \(\displaystyle g\) stands for the Chevalley (root space) generators of non-zero weight.
The generator \(\displaystyle h_i\) is the element of the Cartan subalgebra dual to the
i^th simple root, that is, \(\displaystyle [h_i, g] =\langle \alpha_i , \gamma\rangle g\),
where g is a Chevalley generator, \(\displaystyle \gamma\) is its weight, and
\(\displaystyle \alpha_i\) is the i^th simple root.
The Lie bracket table is too large to be rendered in LaTeX, displaying in html format instead.
roots simple coords epsilon coordinates[,]g_{-36}g_{-35}g_{-34}g_{-33}g_{-32}g_{-31}g_{-30}g_{-29}g_{-28}g_{-27}g_{-26}g_{-25}g_{-24}g_{-23}g_{-22}g_{-21}g_{-20}g_{-19}g_{-18}g_{-17}g_{-16}g_{-15}g_{-14}g_{-13}g_{-12}g_{-11}g_{-10}g_{-9}g_{-8}g_{-7}g_{-6}g_{-5}g_{-4}g_{-3}g_{-2}g_{-1}h_{1}h_{2}h_{3}h_{4}h_{5}h_{6}g_{1}g_{2}g_{3}g_{4}g_{5}g_{6}g_{7}g_{8}g_{9}g_{10}g_{11}g_{12}g_{13}g_{14}g_{15}g_{16}g_{17}g_{18}g_{19}g_{20}g_{21}g_{22}g_{23}g_{24}g_{25}g_{26}g_{27}g_{28}g_{29}g_{30}g_{31}g_{32}g_{33}g_{34}g_{35}g_{36}
(-2, -2, -2, -2, -2, -1)-2e_{1}g_{-36}000000000000000000000000000000000000g_{-36}00000g_{-35}00000g_{-33}0000g_{-31}0000g_{-28}000g_{-25}000-g_{-21}00-g_{-17}00-g_{-12}0-g_{-7}0-g_{-1}-h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}
(-1, -2, -2, -2, -2, -1)-e_{1}-e_{2}g_{-35}000000000000000000000000000000000002g_{-36}01/2g_{-35}00002g_{-34}g_{-33}0000g_{-32}g_{-31}000g_{-29}g_{-28}000g_{-26}g_{-25}00g_{-22}-g_{-21}00-g_{-18}-g_{-17}0-g_{-13}-g_{-12}0-g_{-8}-g_{-7}-g_{-2}-g_{-1}-2h_{6}-4h_{5}-4h_{4}-4h_{3}-4h_{2}-2h_{1}-g_{1}
(0, -2, -2, -2, -2, -1)-2e_{2}g_{-34}00000000000000000000000000000000000g_{-35}-g_{-34}g_{-34}00000g_{-32}00000g_{-29}0000g_{-26}0000g_{-22}000-g_{-18}000-g_{-13}00-g_{-8}00-g_{-2}0-h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-g_{1}0
(-1, -1, -2, -2, -2, -1)-e_{1}-e_{3}g_{-33}000000000000000000000000000002g_{-36}0000g_{-35}01/2g_{-33}-1/2g_{-33}1/2g_{-33}000g_{-32}0g_{-31}0002g_{-30}0g_{-28}00g_{-27}0g_{-25}00g_{-23}0-g_{-21}0g_{-19}0-g_{-17}0-g_{-14}0-g_{-12}-g_{-9}0-g_{-7}-g_{-3}-g_{-1}-2h_{6}-4h_{5}-4h_{4}-4h_{3}-2h_{2}-2h_{1}0-g_{2}-g_{7}
(0, -1, -2, -2, -2, -1)-e_{2}-e_{3}g_{-32}00000000000000000000000000000g_{-35}00002g_{-34}g_{-33}-1/2g_{-32}01/2g_{-32}00002g_{-30}g_{-29}0000g_{-27}g_{-26}000g_{-23}g_{-22}000g_{-19}-g_{-18}00-g_{-14}-g_{-13}00-g_{-9}-g_{-8}0-g_{-3}-g_{-2}0-2h_{6}-4h_{5}-4h_{4}-4h_{3}-2h_{2}-g_{1}-g_{2}-g_{7}0
(-1, -1, -1, -2, -2, -1)-e_{1}-e_{4}g_{-31}0000000000000000000000002g_{-36}000g_{-35}0000g_{-33}001/2g_{-31}0-1/2g_{-31}1/2g_{-31}00g_{-29}00g_{-28}00g_{-27}00g_{-25}02g_{-24}00-g_{-21}0g_{-20}00-g_{-17}g_{-15}00-g_{-12}-g_{-10}0-g_{-7}-g_{-4}-g_{-1}0-2h_{6}-4h_{5}-4h_{4}-2h_{3}-2h_{2}-2h_{1}0-g_{3}0-g_{8}-g_{12}
(0, 0, -2, -2, -2, -1)-2e_{3}g_{-30}00000000000000000000000000000g_{-33}0000g_{-32}00-g_{-30}g_{-30}00000g_{-27}00000g_{-23}0000g_{-19}0000-g_{-14}000-g_{-9}000-g_{-3}00-h_{6}-2h_{5}-2h_{4}-2h_{3}0-g_{2}-g_{7}000
(0, -1, -1, -2, -2, -1)-e_{2}-e_{4}g_{-29}000000000000000000000000g_{-35}0002g_{-34}0000g_{-32}0g_{-31}-1/2g_{-29}1/2g_{-29}-1/2g_{-29}1/2g_{-29}000g_{-27}0g_{-26}0002g_{-24}0g_{-22}00g_{-20}0-g_{-18}00g_{-15}0-g_{-13}0-g_{-10}0-g_{-8}0-g_{-4}-g_{-2}0-2h_{6}-4h_{5}-4h_{4}-2h_{3}-2h_{2}0-g_{1}-g_{3}0-g_{8}-g_{12}0
(-1, -1, -1, -1, -2, -1)-e_{1}-e_{5}g_{-28}00000000000000000002g_{-36}000g_{-35}000g_{-33}0000g_{-31}0001/2g_{-28}00-1/2g_{-28}1/2g_{-28}0g_{-26}000g_{-25}0g_{-23}000-g_{-21}g_{-20}000-g_{-17}2g_{-16}00-g_{-12}g_{-11}0-g_{-7}0-g_{-5}-g_{-1}0-2h_{6}-4h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}00-g_{4}0-g_{9}0-g_{13}-g_{17}
(0, 0, -1, -2, -2, -1)-e_{3}-e_{4}g_{-27}000000000000000000000000g_{-33}000g_{-32}g_{-31}0002g_{-30}g_{-29}00-1/2g_{-27}01/2g_{-27}00002g_{-24}g_{-23}0000g_{-20}g_{-19}000g_{-15}-g_{-14}000-g_{-10}-g_{-9}00-g_{-4}-g_{-3}00-2h_{6}-4h_{5}-4h_{4}-2h_{3}0-g_{2}-g_{3}-g_{7}-g_{8}-g_{12}000
(0, -1, -1, -1, -2, -1)-e_{2}-e_{5}g_{-26}0000000000000000000g_{-35}0002g_{-34}000g_{-32}0000g_{-29}00g_{-28}-1/2g_{-26}1/2g_{-26}0-1/2g_{-26}1/2g_{-26}00g_{-23}00g_{-22}00g_{-20}00-g_{-18}02g_{-16}00-g_{-13}0g_{-11}0-g_{-8}0-g_{-5}-g_{-2}00-2h_{6}-4h_{5}-2h_{4}-2h_{3}-2h_{2}0-g_{1}-g_{4}00-g_{9}0-g_{13}-g_{17}0
(-1, -1, -1, -1, -1, -1)-e_{1}-e_{6}g_{-25}0000000000000002g_{-36}00g_{-35}000g_{-33}000g_{-31}0000g_{-28}00001/2g_{-25}000-1/2g_{-25}g_{-25}g_{-22}0000-g_{-21}g_{-19}000-g_{-17}g_{-15}00-g_{-12}0g_{-11}0-g_{-7}02g_{-6}-g_{-1}00-2h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}00-g_{5}00-g_{10}0-g_{14}0-g_{18}-g_{21}
(0, 0, 0, -2, -2, -1)-2e_{4}g_{-24}000000000000000000000000g_{-31}000g_{-29}0000g_{-27}0000-g_{-24}g_{-24}00000g_{-20}00000g_{-15}0000-g_{-10}0000-g_{-4}000-h_{6}-2h_{5}-2h_{4}00-g_{3}0-g_{8}0-g_{12}00000
(0, 0, -1, -1, -2, -1)-e_{3}-e_{5}g_{-23}0000000000000000000g_{-33}000g_{-32}0002g_{-30}0g_{-28}00g_{-27}0g_{-26}00-1/2g_{-23}1/2g_{-23}-1/2g_{-23}1/2g_{-23}000g_{-20}0g_{-19}0002g_{-16}0-g_{-14}00g_{-11}0-g_{-9}00-g_{-5}-g_{-3}00-2h_{6}-4h_{5}-2h_{4}-2h_{3}00-g_{2}-g_{4}-g_{7}0-g_{9}0-g_{13}-g_{17}000
(0, -1, -1, -1, -1, -1)-e_{2}-e_{6}g_{-22}000000000000000g_{-35}002g_{-34}000g_{-32}000g_{-29}0000g_{-26}000g_{-25}-1/2g_{-22}1/2g_{-22}00-1/2g_{-22}g_{-22}0g_{-19}000-g_{-18}0g_{-15}00-g_{-13}0g_{-11}0-g_{-8}002g_{-6}-g_{-2}00-2h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}00-g_{1}-g_{5}00-g_{10}00-g_{14}0-g_{18}-g_{21}0
(-1, -1, -1, -1, -1, 0)-e_{1}+e_{6}g_{-21}00000000000-2g_{-36}00-g_{-35}00-g_{-33}000-g_{-31}000-g_{-28}0000-g_{-25}000001/2g_{-21}0001/2g_{-21}-g_{-21}g_{-18}000-g_{-17}0g_{-14}00-g_{-12}0g_{-10}0-g_{-7}00g_{-5}-g_{-1}00-2h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}0002g_{6}00g_{11}00g_{15}0g_{19}0g_{22}g_{25}
(0, 0, 0, -1, -2, -1)-e_{4}-e_{5}g_{-20}0000000000000000000g_{-31}000g_{-29}g_{-28}00g_{-27}g_{-26}0002g_{-24}g_{-23}0000-1/2g_{-20}01/2g_{-20}00002g_{-16}g_{-15}0000g_{-11}-g_{-10}000-g_{-5}-g_{-4}000-2h_{6}-4h_{5}-2h_{4}00-g_{3}-g_{4}0-g_{8}-g_{9}-g_{12}-g_{13}0-g_{17}00000
(0, 0, -1, -1, -1, -1)-e_{3}-e_{6}g_{-19}000000000000000g_{-33}00g_{-32}0002g_{-30}000g_{-27}00g_{-25}0g_{-23}00g_{-22}00-1/2g_{-19}1/2g_{-19}0-1/2g_{-19}g_{-19}00g_{-15}00-g_{-14}00g_{-11}0-g_{-9}002g_{-6}-g_{-3}000-2h_{6}-2h_{5}-2h_{4}-2h_{3}00-g_{2}-g_{5}0-g_{7}0-g_{10}00-g_{14}0-g_{18}-g_{21}000
(0, -1, -1, -1, -1, 0)-e_{2}+e_{6}g_{-18}00000000000-g_{-35}00-2g_{-34}00-g_{-32}000-g_{-29}000-g_{-26}0000-g_{-22}0000g_{-21}-1/2g_{-18}1/2g_{-18}001/2g_{-18}-g_{-18}0g_{-14}00-g_{-13}00g_{-10}0-g_{-8}00g_{-5}-g_{-2}000-2h_{5}-2h_{4}-2h_{3}-2h_{2}00-g_{1}2g_{6}000g_{11}00g_{15}00g_{19}0g_{22}g_{25}0
(-1, -1, -1, -1, 0, 0)-e_{1}+e_{5}g_{-17}00000000-2g_{-36}0-g_{-35}00-g_{-33}00-g_{-31}000-g_{-28}0000-g_{-25}00000-g_{-21}00001/2g_{-17}001/2g_{-17}-1/2g_{-17}0g_{-13}00-g_{-12}00g_{-9}0-g_{-7}00g_{-4}-g_{-1}000-2h_{4}-2h_{3}-2h_{2}-2h_{1}000g_{5}000g_{11}002g_{16}00g_{20}0g_{23}0g_{26}g_{28}
(0, 0, 0, 0, -2, -1)-2e_{5}g_{-16}0000000000000000000g_{-28}000g_{-26}000g_{-23}0000g_{-20}000000-g_{-16}g_{-16}00000g_{-11}00000-g_{-5}0000-h_{6}-2h_{5}000-g_{4}00-g_{9}00-g_{13}0-g_{17}00000000
(0, 0, 0, -1, -1, -1)-e_{4}-e_{6}g_{-15}000000000000000g_{-31}00g_{-29}000g_{-27}0g_{-25}02g_{-24}0g_{-22}00g_{-20}0g_{-19}0000-1/2g_{-15}1/2g_{-15}-1/2g_{-15}g_{-15}000g_{-11}0-g_{-10}0002g_{-6}-g_{-4}000-2h_{6}-2h_{5}-2h_{4}000-g_{3}-g_{5}0-g_{8}0-g_{10}-g_{12}0-g_{14}0-g_{18}0-g_{21}00000
(0, 0, -1, -1, -1, 0)-e_{3}+e_{6}g_{-14}00000000000-g_{-33}00-g_{-32}00-2g_{-30}000-g_{-27}000-g_{-23}000g_{-21}-g_{-19}000g_{-18}00-1/2g_{-14}1/2g_{-14}01/2g_{-14}-g_{-14}00g_{-10}0-g_{-9}000g_{-5}-g_{-3}000-2h_{5}-2h_{4}-2h_{3}000-g_{2}2g_{6}0-g_{7}0g_{11}000g_{15}00g_{19}0g_{22}g_{25}000
(0, -1, -1, -1, 0, 0)-e_{2}+e_{5}g_{-13}00000000-g_{-35}0-2g_{-34}00-g_{-32}00-g_{-29}000-g_{-26}0000-g_{-22}00000-g_{-18}000g_{-17}-1/2g_{-13}1/2g_{-13}01/2g_{-13}-1/2g_{-13}00g_{-9}0-g_{-8}000g_{-4}-g_{-2}000-2h_{4}-2h_{3}-2h_{2}000-g_{1}g_{5}000g_{11}0002g_{16}00g_{20}00g_{23}0g_{26}g_{28}0
(-1, -1, -1, 0, 0, 0)-e_{1}+e_{4}g_{-12}00000-2g_{-36}0-g_{-35}0-g_{-33}00-g_{-31}000-g_{-28}0000-g_{-25}0000-g_{-21}00000-g_{-17}0001/2g_{-12}01/2g_{-12}-1/2g_{-12}00g_{-8}0-g_{-7}000g_{-3}-g_{-1}000-2h_{3}-2h_{2}-2h_{1}0000g_{4}000g_{10}000g_{15}00g_{20}002g_{24}0g_{27}0g_{29}g_{31}
(0, 0, 0, 0, -1, -1)-e_{5}-e_{6}g_{-11}000000000000000g_{-28}00g_{-26}g_{-25}00g_{-23}g_{-22}00g_{-20}g_{-19}0002g_{-16}g_{-15}000000-1/2g_{-11}0g_{-11}00002g_{-6}-g_{-5}0000-2h_{6}-2h_{5}000-g_{4}-g_{5}00-g_{9}-g_{10}0-g_{13}-g_{14}0-g_{17}-g_{18}0-g_{21}00000000
(0, 0, 0, -1, -1, 0)-e_{4}+e_{6}g_{-10}00000000000-g_{-31}00-g_{-29}00-g_{-27}000-2g_{-24}00g_{-21}-g_{-20}00g_{-18}0-g_{-15}00g_{-14}0000-1/2g_{-10}1/2g_{-10}1/2g_{-10}-g_{-10}000g_{-5}-g_{-4}0000-2h_{5}-2h_{4}000-g_{3}2g_{6}00-g_{8}0g_{11}-g_{12}00g_{15}00g_{19}0g_{22}0g_{25}00000
(0, 0, -1, -1, 0, 0)-e_{3}+e_{5}g_{-9}00000000-g_{-33}0-g_{-32}00-2g_{-30}00-g_{-27}000-g_{-23}0000-g_{-19}000g_{-17}0-g_{-14}00g_{-13}00-1/2g_{-9}1/2g_{-9}1/2g_{-9}-1/2g_{-9}000g_{-4}-g_{-3}0000-2h_{4}-2h_{3}000-g_{2}g_{5}00-g_{7}0g_{11}0002g_{16}000g_{20}00g_{23}0g_{26}g_{28}000
(0, -1, -1, 0, 0, 0)-e_{2}+e_{4}g_{-8}00000-g_{-35}0-2g_{-34}0-g_{-32}00-g_{-29}000-g_{-26}0000-g_{-22}0000-g_{-18}00000-g_{-13}00g_{-12}-1/2g_{-8}1/2g_{-8}1/2g_{-8}-1/2g_{-8}000g_{-3}-g_{-2}0000-2h_{3}-2h_{2}000-g_{1}g_{4}0000g_{10}000g_{15}000g_{20}002g_{24}00g_{27}0g_{29}g_{31}0
(-1, -1, 0, 0, 0, 0)-e_{1}+e_{3}g_{-7}000-2g_{-36}-g_{-35}0-g_{-33}00-g_{-31}000-g_{-28}000-g_{-25}0000-g_{-21}0000-g_{-17}00000-g_{-12}001/2g_{-7}1/2g_{-7}-1/2g_{-7}000g_{-2}-g_{-1}0000-2h_{2}-2h_{1}0000g_{3}0000g_{9}000g_{14}000g_{19}00g_{23}00g_{27}02g_{30}0g_{32}g_{33}
(0, 0, 0, 0, 0, -1)-2e_{6}g_{-6}000000000000000g_{-25}00g_{-22}000g_{-19}000g_{-15}0000g_{-11}00000000-g_{-6}2g_{-6}00000-h_{6}0000-g_{5}000-g_{10}000-g_{14}00-g_{18}00-g_{21}00000000000
(0, 0, 0, 0, -1, 0)-e_{5}+e_{6}g_{-5}00000000000-g_{-28}00-g_{-26}00-g_{-23}0g_{-21}0-g_{-20}0g_{-18}0-2g_{-16}0g_{-14}00-g_{-11}0g_{-10}000000-1/2g_{-5}g_{-5}-g_{-5}0000-2h_{5}0000-g_{4}2g_{6}00-g_{9}0g_{11}0-g_{13}0g_{15}-g_{17}0g_{19}00g_{22}0g_{25}00000000
(0, 0, 0, -1, 0, 0)-e_{4}+e_{5}g_{-4}00000000-g_{-31}0-g_{-29}00-g_{-27}00-2g_{-24}000-g_{-20}000g_{-17}-g_{-15}00g_{-13}00-g_{-10}0g_{-9}0000-1/2g_{-4}g_{-4}-1/2g_{-4}0000-2h_{4}0000-g_{3}g_{5}00-g_{8}0g_{11}0-g_{12}002g_{16}000g_{20}00g_{23}0g_{26}0g_{28}00000
(0, 0, -1, 0, 0, 0)-e_{3}+e_{4}g_{-3}00000-g_{-33}0-g_{-32}0-2g_{-30}00-g_{-27}000-g_{-23}0000-g_{-19}0000-g_{-14}00g_{-12}00-g_{-9}0g_{-8}00-1/2g_{-3}g_{-3}-1/2g_{-3}0000-2h_{3}0000-g_{2}g_{4}00-g_{7}0g_{10}0000g_{15}000g_{20}0002g_{24}00g_{27}0g_{29}g_{31}000
(0, -1, 0, 0, 0, 0)-e_{2}+e_{3}g_{-2}000-g_{-35}-2g_{-34}0-g_{-32}00-g_{-29}000-g_{-26}000-g_{-22}0000-g_{-18}0000-g_{-13}00000-g_{-8}0g_{-7}-1/2g_{-2}g_{-2}-1/2g_{-2}0000-2h_{2}0000-g_{1}g_{3}0000g_{9}0000g_{14}000g_{19}000g_{23}00g_{27}002g_{30}0g_{32}g_{33}0
(-1, 0, 0, 0, 0, 0)-e_{1}+e_{2}g_{-1}0-2g_{-36}-g_{-35}0-g_{-33}00-g_{-31}00-g_{-28}000-g_{-25}000-g_{-21}0000-g_{-17}0000-g_{-12}00000-g_{-7}0g_{-1}-1/2g_{-1}0000-2h_{1}00000g_{2}0000g_{8}0000g_{13}000g_{18}000g_{22}00g_{26}00g_{29}0g_{32}02g_{34}g_{35}
(0, 0, 0, 0, 0, 0)0h_{1}-g_{-36}0g_{-34}-1/2g_{-33}1/2g_{-32}-1/2g_{-31}01/2g_{-29}-1/2g_{-28}01/2g_{-26}-1/2g_{-25}001/2g_{-22}-1/2g_{-21}001/2g_{-18}-1/2g_{-17}0001/2g_{-13}-1/2g_{-12}0001/2g_{-8}-1/2g_{-7}00001/2g_{-2}-g_{-1}000000g_{1}-1/2g_{2}00001/2g_{7}-1/2g_{8}0001/2g_{12}-1/2g_{13}0001/2g_{17}-1/2g_{18}001/2g_{21}-1/2g_{22}001/2g_{25}-1/2g_{26}01/2g_{28}-1/2g_{29}01/2g_{31}-1/2g_{32}1/2g_{33}-g_{34}0g_{36}
(0, 0, 0, 0, 0, 0)0h_{2}0-1/2g_{-35}-g_{-34}1/2g_{-33}00g_{-30}-1/2g_{-29}01/2g_{-27}-1/2g_{-26}001/2g_{-23}-1/2g_{-22}001/2g_{-19}-1/2g_{-18}0001/2g_{-14}-1/2g_{-13}0001/2g_{-9}-1/2g_{-8}-1/2g_{-7}0001/2g_{-3}-g_{-2}1/2g_{-1}000000-1/2g_{1}g_{2}-1/2g_{3}0001/2g_{7}1/2g_{8}-1/2g_{9}0001/2g_{13}-1/2g_{14}0001/2g_{18}-1/2g_{19}001/2g_{22}-1/2g_{23}001/2g_{26}-1/2g_{27}01/2g_{29}-g_{30}00-1/2g_{33}g_{34}1/2g_{35}0
(0, 0, 0, 0, 0, 0)0h_{3}000-1/2g_{-33}-1/2g_{-32}1/2g_{-31}-g_{-30}1/2g_{-29}0000g_{-24}-1/2g_{-23}001/2g_{-20}-1/2g_{-19}0001/2g_{-15}-1/2g_{-14}0-1/2g_{-12}01/2g_{-10}-1/2g_{-9}-1/2g_{-8}1/2g_{-7}001/2g_{-4}-g_{-3}1/2g_{-2}00000000-1/2g_{2}g_{3}-1/2g_{4}00-1/2g_{7}1/2g_{8}1/2g_{9}-1/2g_{10}01/2g_{12}01/2g_{14}-1/2g_{15}0001/2g_{19}-1/2g_{20}001/2g_{23}-g_{24}0000-1/2g_{29}g_{30}-1/2g_{31}1/2g_{32}1/2g_{33}000
(0, 0, 0, 0, 0, 0)0h_{4}00000-1/2g_{-31}0-1/2g_{-29}1/2g_{-28}-1/2g_{-27}1/2g_{-26}0-g_{-24}1/2g_{-23}00000-1/2g_{-17}g_{-16}-1/2g_{-15}0-1/2g_{-13}1/2g_{-12}1/2g_{-11}-1/2g_{-10}-1/2g_{-9}1/2g_{-8}001/2g_{-5}-g_{-4}1/2g_{-3}0000000000-1/2g_{3}g_{4}-1/2g_{5}00-1/2g_{8}1/2g_{9}1/2g_{10}-1/2g_{11}-1/2g_{12}1/2g_{13}01/2g_{15}-g_{16}1/2g_{17}00000-1/2g_{23}g_{24}0-1/2g_{26}1/2g_{27}-1/2g_{28}1/2g_{29}01/2g_{31}00000
(0, 0, 0, 0, 0, 0)0h_{5}00000000-1/2g_{-28}0-1/2g_{-26}1/2g_{-25}0-1/2g_{-23}1/2g_{-22}-1/2g_{-21}-1/2g_{-20}1/2g_{-19}-1/2g_{-18}1/2g_{-17}-g_{-16}1/2g_{-15}-1/2g_{-14}1/2g_{-13}00-1/2g_{-10}1/2g_{-9}00g_{-6}-g_{-5}1/2g_{-4}000000000000-1/2g_{4}g_{5}-g_{6}00-1/2g_{9}1/2g_{10}00-1/2g_{13}1/2g_{14}-1/2g_{15}g_{16}-1/2g_{17}1/2g_{18}-1/2g_{19}1/2g_{20}1/2g_{21}-1/2g_{22}1/2g_{23}0-1/2g_{25}1/2g_{26}01/2g_{28}00000000
(0, 0, 0, 0, 0, 0)0h_{6}00000000000-g_{-25}00-g_{-22}g_{-21}0-g_{-19}g_{-18}00-g_{-15}g_{-14}00-g_{-11}g_{-10}000-2g_{-6}g_{-5}00000000000000-g_{5}2g_{6}000-g_{10}g_{11}00-g_{14}g_{15}00-g_{18}g_{19}0-g_{21}g_{22}00g_{25}00000000000
(1, 0, 0, 0, 0, 0)e_{1}-e_{2}g_{1}-g_{-35}-2g_{-34}0-g_{-32}0-g_{-29}00-g_{-26}00-g_{-22}000-g_{-18}000-g_{-13}0000-g_{-8}0000-g_{-2}000002h_{1}-g_{1}1/2g_{1}00000g_{7}00000g_{12}0000g_{17}0000g_{21}000g_{25}000g_{28}00g_{31}00g_{33}0g_{35}2g_{36}0
(0, 1, 0, 0, 0, 0)e_{2}-e_{3}g_{2}0-g_{-33}-g_{-32}0-2g_{-30}00-g_{-27}00-g_{-23}000-g_{-19}000-g_{-14}0000-g_{-9}0000-g_{-3}g_{-1}00002h_{2}01/2g_{2}-g_{2}1/2g_{2}000-g_{7}0g_{8}00000g_{13}0000g_{18}0000g_{22}000g_{26}000g_{29}00g_{32}02g_{34}g_{35}000
(0, 0, 1, 0, 0, 0)e_{3}-e_{4}g_{3}000-g_{-31}-g_{-29}0-g_{-27}00-2g_{-24}000-g_{-20}000-g_{-15}0000-g_{-10}0g_{-7}00-g_{-4}g_{-2}00002h_{3}0001/2g_{3}-g_{3}1/2g_{3}000-g_{8}0g_{9}00-g_{12}00g_{14}0000g_{19}0000g_{23}000g_{27}002g_{30}0g_{32}0g_{33}00000
(0, 0, 0, 1, 0, 0)e_{4}-e_{5}g_{4}00000-g_{-28}0-g_{-26}0-g_{-23}00-g_{-20}000-2g_{-16}00g_{-12}0-g_{-11}0g_{-8}00-g_{-5}g_{-3}00002h_{4}000001/2g_{4}-g_{4}1/2g_{4}000-g_{9}0g_{10}00-g_{13}00g_{15}-g_{17}000g_{20}0002g_{24}00g_{27}00g_{29}0g_{31}00000000
(0, 0, 0, 0, 1, 0)e_{5}-e_{6}g_{5}00000000-g_{-25}0-g_{-22}00-g_{-19}0g_{-17}-g_{-15}0g_{-13}0-g_{-11}0g_{-9}00-2g_{-6}g_{-4}00002h_{5}00000001/2g_{5}-g_{5}g_{5}000-g_{10}0g_{11}00-g_{14}02g_{16}0-g_{18}0g_{20}0-g_{21}0g_{23}00g_{26}00g_{28}00000000000
(0, 0, 0, 0, 0, 1)2e_{6}g_{6}00000000000g_{-21}00g_{-18}00g_{-14}000g_{-10}000g_{-5}0000h_{6}000000000g_{6}-2g_{6}0000-g_{11}0000-g_{15}000-g_{19}000-g_{22}00-g_{25}000000000000000
(1, 1, 0, 0, 0, 0)e_{1}-e_{3}g_{7}-g_{-33}-g_{-32}0-2g_{-30}0-g_{-27}00-g_{-23}00-g_{-19}000-g_{-14}000-g_{-9}0000-g_{-3}00002h_{2}+2h_{1}0000g_{1}-g_{2}-1/2g_{7}-1/2g_{7}1/2g_{7}00000g_{12}00000g_{17}0000g_{21}0000g_{25}000g_{28}000g_{31}00g_{33}0g_{35}2g_{36}000
(0, 1, 1, 0, 0, 0)e_{2}-e_{4}g_{8}0-g_{-31}-g_{-29}0-g_{-27}00-2g_{-24}00-g_{-20}000-g_{-15}000-g_{-10}0000-g_{-4}g_{-1}0002h_{3}+2h_{2}0000g_{2}-g_{3}01/2g_{8}-1/2g_{8}-1/2g_{8}1/2g_{8}00-g_{12}00g_{13}00000g_{18}0000g_{22}0000g_{26}000g_{29}00g_{32}02g_{34}0g_{35}00000
(0, 0, 1, 1, 0, 0)e_{3}-e_{5}g_{9}000-g_{-28}-g_{-26}0-g_{-23}00-g_{-20}000-2g_{-16}000-g_{-11}0g_{-7}00-g_{-5}g_{-2}0002h_{4}+2h_{3}0000g_{3}-g_{4}0001/2g_{9}-1/2g_{9}-1/2g_{9}1/2g_{9}00-g_{13}00g_{14}0-g_{17}000g_{19}0000g_{23}000g_{27}002g_{30}00g_{32}0g_{33}00000000
(0, 0, 0, 1, 1, 0)e_{4}-e_{6}g_{10}00000-g_{-25}0-g_{-22}0-g_{-19}00-g_{-15}00g_{-12}-g_{-11}0g_{-8}00-2g_{-6}g_{-3}0002h_{5}+2h_{4}0000g_{4}-g_{5}000001/2g_{10}-1/2g_{10}-1/2g_{10}g_{10}00-g_{14}00g_{15}0-g_{18}00g_{20}-g_{21}002g_{24}000g_{27}00g_{29}00g_{31}00000000000
(0, 0, 0, 0, 1, 1)e_{5}+e_{6}g_{11}00000000g_{-21}0g_{-18}g_{-17}0g_{-14}g_{-13}0g_{-10}g_{-9}00g_{-5}g_{-4}0002h_{6}+2h_{5}0000g_{5}-2g_{6}00000001/2g_{11}0-g_{11}000-g_{15}-2g_{16}000-g_{19}-g_{20}00-g_{22}-g_{23}00-g_{25}-g_{26}00-g_{28}000000000000000
(1, 1, 1, 0, 0, 0)e_{1}-e_{4}g_{12}-g_{-31}-g_{-29}0-g_{-27}0-2g_{-24}00-g_{-20}00-g_{-15}000-g_{-10}000-g_{-4}00002h_{3}+2h_{2}+2h_{1}000g_{1}-g_{3}000g_{7}0-g_{8}-1/2g_{12}0-1/2g_{12}1/2g_{12}00000g_{17}00000g_{21}0000g_{25}0000g_{28}000g_{31}00g_{33}0g_{35}02g_{36}00000
(0, 1, 1, 1, 0, 0)e_{2}-e_{5}g_{13}0-g_{-28}-g_{-26}0-g_{-23}00-g_{-20}00-2g_{-16}000-g_{-11}000-g_{-5}g_{-1}0002h_{4}+2h_{3}+2h_{2}000g_{2}-g_{4}000g_{8}0-g_{9}01/2g_{13}-1/2g_{13}0-1/2g_{13}1/2g_{13}0-g_{17}000g_{18}00000g_{22}0000g_{26}000g_{29}00g_{32}002g_{34}0g_{35}00000000
(0, 0, 1, 1, 1, 0)e_{3}-e_{6}g_{14}000-g_{-25}-g_{-22}0-g_{-19}00-g_{-15}000-g_{-11}0g_{-7}0-2g_{-6}g_{-2}0002h_{5}+2h_{4}+2h_{3}000g_{3}-g_{5}000g_{9}0-g_{10}0001/2g_{14}-1/2g_{14}0-1/2g_{14}g_{14}0-g_{18}000g_{19}-g_{21}000g_{23}000g_{27}0002g_{30}00g_{32}00g_{33}00000000000
(0, 0, 0, 1, 1, 1)e_{4}+e_{6}g_{15}00000g_{-21}0g_{-18}0g_{-14}0g_{-12}g_{-10}0g_{-8}0g_{-5}g_{-3}0002h_{6}+2h_{5}+2h_{4}000g_{4}-2g_{6}000g_{10}0-g_{11}000001/2g_{15}-1/2g_{15}1/2g_{15}-g_{15}00-g_{19}0-g_{20}00-g_{22}0-2g_{24}0-g_{25}0-g_{27}000-g_{29}00-g_{31}000000000000000
(0, 0, 0, 0, 2, 1)2e_{5}g_{16}00000000g_{-17}0g_{-13}00g_{-9}00g_{-4}000h_{6}+2h_{5}0000g_{5}00000-g_{11}0000000g_{16}-g_{16}0000-g_{20}0000-g_{23}000-g_{26}000-g_{28}0000000000000000000
(1, 1, 1, 1, 0, 0)e_{1}-e_{5}g_{17}-g_{-28}-g_{-26}0-g_{-23}0-g_{-20}00-2g_{-16}00-g_{-11}000-g_{-5}0002h_{4}+2h_{3}+2h_{2}+2h_{1}000g_{1}-g_{4}00g_{7}0-g_{9}00g_{12}00-g_{13}-1/2g_{17}00-1/2g_{17}1/2g_{17}00000g_{21}00000g_{25}0000g_{28}000g_{31}00g_{33}00g_{35}02g_{36}00000000
(0, 1, 1, 1, 1, 0)e_{2}-e_{6}g_{18}0-g_{-25}-g_{-22}0-g_{-19}00-g_{-15}00-g_{-11}000-2g_{-6}g_{-1}002h_{5}+2h_{4}+2h_{3}+2h_{2}000g_{2}-g_{5}00g_{8}0-g_{10}00g_{13}00-g_{14}01/2g_{18}-1/2g_{18}00-1/2g_{18}g_{18}-g_{21}0000g_{22}0000g_{26}000g_{29}000g_{32}002g_{34}00g_{35}00000000000
(0, 0, 1, 1, 1, 1)e_{3}+e_{6}g_{19}000g_{-21}g_{-18}0g_{-14}00g_{-10}0g_{-7}0g_{-5}g_{-2}002h_{6}+2h_{5}+2h_{4}+2h_{3}000g_{3}-2g_{6}00g_{9}0-g_{11}00g_{14}00-g_{15}0001/2g_{19}-1/2g_{19}01/2g_{19}-g_{19}0-g_{22}00-g_{23}0-g_{25}00-g_{27}000-2g_{30}000-g_{32}00-g_{33}000000000000000
(0, 0, 0, 1, 2, 1)e_{4}+e_{5}g_{20}00000g_{-17}0g_{-13}g_{-12}g_{-9}g_{-8}0g_{-4}g_{-3}002h_{6}+4h_{5}+2h_{4}000g_{4}g_{5}000g_{10}-g_{11}0000-g_{15}-2g_{16}000001/2g_{20}0-1/2g_{20}000-g_{23}-2g_{24}000-g_{26}-g_{27}00-g_{28}-g_{29}000-g_{31}0000000000000000000
(1, 1, 1, 1, 1, 0)e_{1}-e_{6}g_{21}-g_{-25}-g_{-22}0-g_{-19}0-g_{-15}00-g_{-11}00-2g_{-6}0002h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}00g_{1}-g_{5}00g_{7}0-g_{10}0g_{12}00-g_{14}0g_{17}000-g_{18}-1/2g_{21}000-1/2g_{21}g_{21}00000g_{25}0000g_{28}000g_{31}000g_{33}00g_{35}002g_{36}00000000000
(0, 1, 1, 1, 1, 1)e_{2}+e_{6}g_{22}0g_{-21}g_{-18}0g_{-14}00g_{-10}00g_{-5}g_{-1}002h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}00g_{2}-2g_{6}00g_{8}0-g_{11}0g_{13}00-g_{15}0g_{18}000-g_{19}01/2g_{22}-1/2g_{22}001/2g_{22}-g_{22}-g_{25}000-g_{26}0000-g_{29}000-g_{32}000-2g_{34}00-g_{35}000000000000000
(0, 0, 1, 1, 2, 1)e_{3}+e_{5}g_{23}000g_{-17}g_{-13}0g_{-9}0g_{-7}g_{-4}g_{-2}002h_{6}+4h_{5}+2h_{4}+2h_{3}00g_{3}g_{5}00g_{9}0-g_{11}00g_{14}0-2g_{16}000-g_{19}0-g_{20}0001/2g_{23}-1/2g_{23}1/2g_{23}-1/2g_{23}00-g_{26}0-g_{27}00-g_{28}0-2g_{30}000-g_{32}000-g_{33}0000000000000000000
(0, 0, 0, 2, 2, 1)2e_{4}g_{24}00000g_{-12}0g_{-8}0g_{-3}00h_{6}+2h_{5}+2h_{4}000g_{4}0000g_{10}0000-g_{15}00000-g_{20}00000g_{24}-g_{24}0000-g_{27}0000-g_{29}000-g_{31}000000000000000000000000
(1, 1, 1, 1, 1, 1)e_{1}+e_{6}g_{25}g_{-21}g_{-18}0g_{-14}0g_{-10}00g_{-5}002h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}00g_{1}-2g_{6}0g_{7}0-g_{11}0g_{12}00-g_{15}g_{17}000-g_{19}g_{21}0000-g_{22}-1/2g_{25}0001/2g_{25}-g_{25}0000-g_{28}0000-g_{31}000-g_{33}000-g_{35}00-2g_{36}000000000000000
(0, 1, 1, 1, 2, 1)e_{2}+e_{5}g_{26}0g_{-17}g_{-13}0g_{-9}00g_{-4}g_{-1}02h_{6}+4h_{5}+2h_{4}+2h_{3}+2h_{2}00g_{2}g_{5}0g_{8}0-g_{11}0g_{13}00-2g_{16}0g_{18}00-g_{20}00-g_{22}00-g_{23}01/2g_{26}-1/2g_{26}01/2g_{26}-1/2g_{26}0-g_{28}00-g_{29}0000-g_{32}000-2g_{34}000-g_{35}0000000000000000000
(0, 0, 1, 2, 2, 1)e_{3}+e_{4}g_{27}000g_{-12}g_{-8}g_{-7}g_{-3}g_{-2}02h_{6}+4h_{5}+4h_{4}+2h_{3}00g_{3}g_{4}00g_{9}g_{10}000g_{14}-g_{15}000-g_{19}-g_{20}0000-g_{23}-2g_{24}0001/2g_{27}0-1/2g_{27}000-g_{29}-2g_{30}000-g_{31}-g_{32}000-g_{33}000000000000000000000000
(1, 1, 1, 1, 2, 1)e_{1}+e_{5}g_{28}g_{-17}g_{-13}0g_{-9}0g_{-4}002h_{6}+4h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}g_{5}0g_{7}0-g_{11}g_{12}00-2g_{16}g_{17}000-g_{20}g_{21}000-g_{23}0-g_{25}000-g_{26}-1/2g_{28}001/2g_{28}-1/2g_{28}0000-g_{31}0000-g_{33}000-g_{35}000-2g_{36}0000000000000000000
(0, 1, 1, 2, 2, 1)e_{2}+e_{4}g_{29}0g_{-12}g_{-8}0g_{-3}g_{-1}02h_{6}+4h_{5}+4h_{4}+2h_{3}+2h_{2}0g_{2}g_{4}0g_{8}0g_{10}0g_{13}0-g_{15}00g_{18}0-g_{20}00-g_{22}0-2g_{24}000-g_{26}0-g_{27}01/2g_{29}-1/2g_{29}1/2g_{29}-1/2g_{29}00-g_{31}0-g_{32}0000-2g_{34}000-g_{35}000000000000000000000000
(0, 0, 2, 2, 2, 1)2e_{3}g_{30}000g_{-7}g_{-2}0h_{6}+2h_{5}+2h_{4}+2h_{3}00g_{3}000g_{9}000g_{14}0000-g_{19}0000-g_{23}00000-g_{27}000g_{30}-g_{30}0000-g_{32}0000-g_{33}00000000000000000000000000000
(1, 1, 1, 2, 2, 1)e_{1}+e_{4}g_{31}g_{-12}g_{-8}0g_{-3}02h_{6}+4h_{5}+4h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}g_{4}g_{7}0g_{10}g_{12}00-g_{15}g_{17}00-g_{20}0g_{21}00-2g_{24}0-g_{25}00-g_{27}00-g_{28}00-g_{29}-1/2g_{31}01/2g_{31}-1/2g_{31}0000-g_{33}0000-g_{35}000-2g_{36}000000000000000000000000
(0, 1, 2, 2, 2, 1)e_{2}+e_{3}g_{32}0g_{-7}g_{-2}g_{-1}2h_{6}+4h_{5}+4h_{4}+4h_{3}+2h_{2}0g_{2}g_{3}0g_{8}g_{9}00g_{13}g_{14}00g_{18}-g_{19}000-g_{22}-g_{23}000-g_{26}-g_{27}0000-g_{29}-2g_{30}01/2g_{32}0-1/2g_{32}000-g_{33}-2g_{34}0000-g_{35}00000000000000000000000000000
(1, 1, 2, 2, 2, 1)e_{1}+e_{3}g_{33}g_{-7}g_{-2}02h_{6}+4h_{5}+4h_{4}+4h_{3}+2h_{2}+2h_{1}g_{1}g_{3}g_{7}0g_{9}g_{12}0g_{14}0g_{17}0-g_{19}0g_{21}0-g_{23}00-g_{25}0-g_{27}00-g_{28}0-2g_{30}000-g_{31}0-g_{32}-1/2g_{33}1/2g_{33}-1/2g_{33}0000-g_{35}0000-2g_{36}00000000000000000000000000000
(0, 2, 2, 2, 2, 1)2e_{2}g_{34}0g_{-1}h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}0g_{2}00g_{8}00g_{13}000g_{18}000-g_{22}0000-g_{26}0000-g_{29}00000-g_{32}0g_{34}-g_{34}0000-g_{35}00000000000000000000000000000000000
(1, 2, 2, 2, 2, 1)e_{1}+e_{2}g_{35}g_{-1}2h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}g_{1}g_{2}g_{7}g_{8}0g_{12}g_{13}0g_{17}g_{18}00g_{21}-g_{22}00-g_{25}-g_{26}000-g_{28}-g_{29}000-g_{31}-g_{32}0000-g_{33}-2g_{34}0-1/2g_{35}0000-2g_{36}00000000000000000000000000000000000
(2, 2, 2, 2, 2, 1)2e_{1}g_{36}h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}g_{1}0g_{7}0g_{12}00g_{17}00g_{21}000-g_{25}000-g_{28}0000-g_{31}0000-g_{33}00000-g_{35}-g_{36}00000000000000000000000000000000000000000
We define the symmetric Cartan matrix
by requesting that the entry in the i-th row and j-th column
be the scalar product of the i^th and j^th roots. The symmetric Cartan matrix is:
\(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0\\ 0 & -1/2 & 1 & -1/2 & 0 & 0\\ 0 & 0 & -1/2 & 1 & -1/2 & 0\\ 0 & 0 & 0 & -1/2 & 1 & -1\\ 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Let the (i, j)^{th} entry of the symmetric Cartan matrix be a_{ij}.
Then we define the co-symmetric Cartan matrix as the matrix whose (i, j)^{th} entry equals 4*a_{ij}/(a_{ii}*a_{jj}). In other words, the co-symmetric Cartan matrix is the symmetric Cartan matrix of the dual root system. The co-symmetric Cartan matrix equals:
\(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0 & 0\\ 0 & -2 & 4 & -2 & 0 & 0\\ 0 & 0 & -2 & 4 & -2 & 0\\ 0 & 0 & 0 & -2 & 4 & -2\\ 0 & 0 & 0 & 0 & -2 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1/16
Half sum of positive roots: (6, 11, 15, 18, 20, 21/2)= \(\displaystyle 6\varepsilon_{1}+5\varepsilon_{2}+4\varepsilon_{3}+3\varepsilon_{4}+2\varepsilon_{5}+\varepsilon_{6}\)
The fundamental weights (the j^th fundamental weight has scalar product 1
with the j^th simple root times 2 divided by the root length squared,
and 0 with the remaining simple roots):
(1, 1, 1, 1, 1, 1/2) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 2, 1) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3, 3/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 4, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5, 5/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
(1, 2, 3, 4, 5, 3) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}\)
Below is the simple basis realized in epsilon coordinates. Please note that the epsilon coordinate realizations do not have long roots of length of 2 in types G and C. This means that gramm matrix (w.r.t. the standard scalar product) of the epsilon coordinate realizations in types G and C does not equal the corresponding symmetric Cartan matrix.
(1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1) = \(\displaystyle 2\varepsilon_{6}\)