sl(9), type \(A^{1}_8\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: A^{1}_8.
Weyl group size: 362880.
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.
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The root system has 72 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-1, -1, -1, -1, -1, -1, -1, -1)-e_{1}+e_{9}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, -1, -1, -1, -1, -1, -1, -1)-e_{2}+e_{9}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, -1, -1, 0)-e_{1}+e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, -1, -1, -1, -1, -1, -1)-e_{3}+e_{9}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, -1, -1, 0)-e_{2}+e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, -1, 0, 0)-e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, -1, -1, -1, -1, -1)-e_{4}+e_{9}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, -1, 0)-e_{3}+e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, -1, 0, 0)-e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, 0, 0, 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, 0, -1, -1, -1, -1)-e_{5}+e_{9}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, -1, 0)-e_{4}+e_{8}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, 0, 0)-e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, 0, 0, 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, 0, 0)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, -1, -1)-e_{6}+e_{9}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, -1, 0)-e_{5}+e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, 0, 0)-e_{4}+e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, -1, -1)-e_{7}+e_{9}\(s_{7}s_{8}s_{7}\)
(0, 0, 0, 0, 0, -1, -1, 0)-e_{6}+e_{8}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, 0, 0)-e_{5}+e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, 0, -1)-e_{8}+e_{9}\(s_{8}\)
(0, 0, 0, 0, 0, 0, -1, 0)-e_{7}+e_{8}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0, 0)-e_{6}+e_{7}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0, 0)e_{6}-e_{7}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 0)e_{7}-e_{8}\(s_{7}\)
(0, 0, 0, 0, 0, 0, 0, 1)e_{8}-e_{9}\(s_{8}\)
(1, 1, 0, 0, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0, 0)e_{5}-e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 0)e_{6}-e_{8}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 1)e_{7}-e_{9}\(s_{7}s_{8}s_{7}\)
(1, 1, 1, 0, 0, 0, 0, 0)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 0, 0, 0, 0)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0, 0, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0, 0)e_{4}-e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 0)e_{5}-e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 1)e_{6}-e_{9}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(1, 1, 1, 1, 0, 0, 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, 0, 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, 0, 0)e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 0)e_{4}-e_{8}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 1)e_{5}-e_{9}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 0, 0, 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, 0, 0)e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 0)e_{3}-e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 1)e_{4}-e_{9}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 1, 1, 0, 0)e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 0)e_{2}-e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 1)e_{3}-e_{9}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 1, 1, 1, 1, 1, 0)e_{1}-e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 1)e_{2}-e_{9}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(1, 1, 1, 1, 1, 1, 1, 1)e_{1}-e_{9}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-1, -1, -1, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, -1, 0), (0, 0, -1, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -1, -1, -1, 0, 0), (0, 0, 0, -1, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, -1, -1, -1, -1), (0, 0, 0, -1, -1, -1, -1, 0), (0, 0, -1, -1, -1, -1, 0, 0), (0, -1, -1, -1, -1, 0, 0, 0), (-1, -1, -1, -1, 0, 0, 0, 0), (0, 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, -1, -1, -1, 0, 0, 0), (0, -1, -1, -1, 0, 0, 0, 0), (-1, -1, -1, 0, 0, 0, 0, 0), (0, 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, 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, 0), (0, 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, 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, 0), (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1), (1, 1, 0, 0, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 0, 1, 1), (1, 1, 1, 0, 0, 0, 0, 0), (0, 1, 1, 1, 0, 0, 0, 0), (0, 0, 1, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 0, 0, 0, 0), (0, 1, 1, 1, 1, 0, 0, 0), (0, 0, 1, 1, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 1, 0), (0, 0, 0, 0, 1, 1, 1, 1), (1, 1, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1, 1, 1) The resulting Lie bracket pairing table follows.
Type A^{1}_8.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}h_{7}h_{8}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}
(-1, -1, -1, -1, -1, -1, -1, -1)-e_{1}+e_{9}g_{-36}000000000000000000000000000000000000g_{-36}000000g_{-36}g_{-35}000000-g_{-34}g_{-33}00000-g_{-31}g_{-30}0000-g_{-27}g_{-26}000-g_{-22}g_{-21}00-g_{-16}g_{-15}0-g_{-9}g_{-8}-g_{-1}-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}
(0, -1, -1, -1, -1, -1, -1, -1)-e_{2}+e_{9}g_{-35}00000000000000000000000000000000000g_{-36}-g_{-35}g_{-35}00000g_{-35}0g_{-33}00000-g_{-32}0g_{-30}0000-g_{-28}0g_{-26}000-g_{-23}0g_{-21}00-g_{-17}0g_{-15}0-g_{-10}0g_{-8}-g_{-2}0-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-g_{1}
(-1, -1, -1, -1, -1, -1, -1, 0)-e_{1}+e_{8}g_{-34}0000000000000000000000000000-g_{-36}0000000g_{-34}00000g_{-34}-g_{-34}g_{-32}00000-g_{-31}0g_{-29}0000-g_{-27}0g_{-25}000-g_{-22}0g_{-20}00-g_{-16}0g_{-14}0-g_{-9}0g_{-7}-g_{-1}0-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}0g_{8}
(0, 0, -1, -1, -1, -1, -1, -1)-e_{3}+e_{9}g_{-33}000000000000000000000000000g_{-36}000000g_{-35}00-g_{-33}g_{-33}0000g_{-33}00g_{-30}0000-g_{-29}00g_{-26}000-g_{-24}00g_{-21}00-g_{-18}00g_{-15}0-g_{-11}00g_{-8}-g_{-3}00-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}0-g_{2}-g_{9}
(0, -1, -1, -1, -1, -1, -1, 0)-e_{2}+e_{8}g_{-32}0000000000000000000000000000-g_{-35}000000g_{-34}-g_{-32}g_{-32}0000g_{-32}-g_{-32}0g_{-29}0000-g_{-28}00g_{-25}000-g_{-23}00g_{-20}00-g_{-17}00g_{-14}0-g_{-10}00g_{-7}-g_{-2}00-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}0-g_{1}g_{8}0
(-1, -1, -1, -1, -1, -1, 0, 0)-e_{1}+e_{7}g_{-31}000000000000000000000-g_{-36}0000000-g_{-34}000000g_{-31}0000g_{-31}-g_{-31}0g_{-28}0000-g_{-27}00g_{-24}000-g_{-22}00g_{-19}00-g_{-16}00g_{-13}0-g_{-9}00g_{-6}-g_{-1}00-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}00g_{7}0g_{15}
(0, 0, 0, -1, -1, -1, -1, -1)-e_{4}+e_{9}g_{-30}00000000000000000000g_{-36}00000g_{-35}000000g_{-33}0000-g_{-30}g_{-30}000g_{-30}000g_{-26}000-g_{-25}000g_{-21}00-g_{-19}000g_{-15}0-g_{-12}000g_{-8}-g_{-4}000-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}00-g_{3}0-g_{10}-g_{16}
(0, 0, -1, -1, -1, -1, -1, 0)-e_{3}+e_{8}g_{-29}000000000000000000000000000g_{-34}-g_{-33}00000g_{-32}00-g_{-29}g_{-29}000g_{-29}-g_{-29}00g_{-25}000-g_{-24}000g_{-20}00-g_{-18}000g_{-14}0-g_{-11}000g_{-7}-g_{-3}000-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}00-g_{2}g_{8}-g_{9}00
(0, -1, -1, -1, -1, -1, 0, 0)-e_{2}+e_{7}g_{-28}000000000000000000000-g_{-35}0000000-g_{-32}00000g_{-31}-g_{-28}g_{-28}000g_{-28}-g_{-28}00g_{-24}000-g_{-23}000g_{-19}00-g_{-17}000g_{-13}0-g_{-10}000g_{-6}-g_{-2}000-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}00-g_{1}g_{7}00g_{15}0
(-1, -1, -1, -1, -1, 0, 0, 0)-e_{1}+e_{6}g_{-27}000000000000000-g_{-36}000000-g_{-34}0000000-g_{-31}00000g_{-27}000g_{-27}-g_{-27}00g_{-23}000-g_{-22}000g_{-18}00-g_{-16}000g_{-12}0-g_{-9}000g_{-5}-g_{-1}000-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}000g_{6}00g_{14}0g_{21}
(0, 0, 0, 0, -1, -1, -1, -1)-e_{5}+e_{9}g_{-26}00000000000000g_{-36}0000g_{-35}00000g_{-33}000000g_{-30}000000-g_{-26}g_{-26}00g_{-26}0000g_{-21}00-g_{-20}0000g_{-15}0-g_{-13}0000g_{-8}-g_{-5}0000-h_{8}-h_{7}-h_{6}-h_{5}000-g_{4}00-g_{11}0-g_{17}-g_{22}
(0, 0, 0, -1, -1, -1, -1, 0)-e_{4}+e_{8}g_{-25}00000000000000000000g_{-34}00000g_{-32}0-g_{-30}0000g_{-29}0000-g_{-25}g_{-25}00g_{-25}-g_{-25}000g_{-20}00-g_{-19}0000g_{-14}0-g_{-12}0000g_{-7}-g_{-4}0000-h_{7}-h_{6}-h_{5}-h_{4}000-g_{3}g_{8}0-g_{10}0-g_{16}00
(0, 0, -1, -1, -1, -1, 0, 0)-e_{3}+e_{7}g_{-24}000000000000000000000-g_{-33}00000g_{-31}0-g_{-29}0000g_{-28}00-g_{-24}g_{-24}00g_{-24}-g_{-24}000g_{-19}00-g_{-18}0000g_{-13}0-g_{-11}0000g_{-6}-g_{-3}0000-h_{6}-h_{5}-h_{4}-h_{3}000-g_{2}g_{7}0-g_{9}0g_{15}000
(0, -1, -1, -1, -1, 0, 0, 0)-e_{2}+e_{6}g_{-23}000000000000000-g_{-35}000000-g_{-32}0000000-g_{-28}0000g_{-27}-g_{-23}g_{-23}00g_{-23}-g_{-23}000g_{-18}00-g_{-17}0000g_{-12}0-g_{-10}0000g_{-5}-g_{-2}0000-h_{5}-h_{4}-h_{3}-h_{2}000-g_{1}g_{6}000g_{14}00g_{21}0
(-1, -1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{5}g_{-22}0000000000-g_{-36}00000-g_{-34}000000-g_{-31}0000000-g_{-27}0000g_{-22}00g_{-22}-g_{-22}000g_{-17}00-g_{-16}0000g_{-11}0-g_{-9}0000g_{-4}-g_{-1}0000-h_{4}-h_{3}-h_{2}-h_{1}0000g_{5}000g_{13}00g_{20}0g_{26}
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(1, 1, 1, 1, 1, 0, 0, 0)e_{1}-e_{6}g_{27}-g_{-21}0-g_{-14}00-g_{-6}000h_{5}+h_{4}+h_{3}+h_{2}+h_{1}000g_{1}-g_{5}000g_{9}0-g_{12}000g_{16}00-g_{18}000g_{22}000-g_{23}-g_{27}000-g_{27}g_{27}0000000g_{31}0000000g_{34}000000g_{36}000000000000000
(0, 1, 1, 1, 1, 1, 0, 0)e_{2}-e_{7}g_{28}0-g_{-15}00-g_{-7}g_{-1}00h_{6}+h_{5}+h_{4}+h_{3}+h_{2}000g_{2}-g_{6}000g_{10}0-g_{13}000g_{17}00-g_{19}000g_{23}000-g_{24}0g_{28}-g_{28}000-g_{28}g_{28}0-g_{31}00000g_{32}0000000g_{35}000000000000000000000
(0, 0, 1, 1, 1, 1, 1, 0)e_{3}-e_{8}g_{29}00g_{-9}-g_{-8}g_{-2}00h_{7}+h_{6}+h_{5}+h_{4}+h_{3}000g_{3}-g_{7}000g_{11}0-g_{14}000g_{18}00-g_{20}000g_{24}000-g_{25}000g_{29}-g_{29}000-g_{29}g_{29}0-g_{32}00000g_{33}-g_{34}000000000000000000000000000
(0, 0, 0, 1, 1, 1, 1, 1)e_{4}-e_{9}g_{30}g_{-16}g_{-10}0g_{-3}00h_{8}+h_{7}+h_{6}+h_{5}+h_{4}000g_{4}-g_{8}000g_{12}0-g_{15}000g_{19}00-g_{21}000g_{25}000-g_{26}00000g_{30}-g_{30}000-g_{30}00-g_{33}000000-g_{35}00000-g_{36}00000000000000000000
(1, 1, 1, 1, 1, 1, 0, 0)e_{1}-e_{7}g_{31}-g_{-15}0-g_{-7}00h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}00g_{1}-g_{6}00g_{9}0-g_{13}00g_{16}00-g_{19}00g_{22}000-g_{24}00g_{27}0000-g_{28}-g_{31}0000-g_{31}g_{31}0000000g_{34}0000000g_{36}000000000000000000000
(0, 1, 1, 1, 1, 1, 1, 0)e_{2}-e_{8}g_{32}0-g_{-8}g_{-1}0h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}00g_{2}-g_{7}00g_{10}0-g_{14}00g_{17}00-g_{20}00g_{23}000-g_{25}00g_{28}0000-g_{29}0g_{32}-g_{32}0000-g_{32}g_{32}-g_{34}000000g_{35}0000000000000000000000000000
(0, 0, 1, 1, 1, 1, 1, 1)e_{3}-e_{9}g_{33}g_{-9}g_{-2}0h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}00g_{3}-g_{8}00g_{11}0-g_{15}00g_{18}00-g_{21}00g_{24}000-g_{26}00g_{29}0000-g_{30}000g_{33}-g_{33}0000-g_{33}0-g_{35}000000-g_{36}000000000000000000000000000
(1, 1, 1, 1, 1, 1, 1, 0)e_{1}-e_{8}g_{34}-g_{-8}0h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}-g_{7}0g_{9}0-g_{14}0g_{16}00-g_{20}0g_{22}000-g_{25}0g_{27}0000-g_{29}0g_{31}00000-g_{32}-g_{34}00000-g_{34}g_{34}0000000g_{36}0000000000000000000000000000
(0, 1, 1, 1, 1, 1, 1, 1)e_{2}-e_{9}g_{35}g_{-1}h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}0g_{2}-g_{8}0g_{10}0-g_{15}0g_{17}00-g_{21}0g_{23}000-g_{26}0g_{28}0000-g_{30}0g_{32}00000-g_{33}0g_{35}-g_{35}00000-g_{35}-g_{36}00000000000000000000000000000000000
(1, 1, 1, 1, 1, 1, 1, 1)e_{1}-e_{9}g_{36}h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}g_{1}-g_{8}g_{9}0-g_{15}g_{16}00-g_{21}g_{22}000-g_{26}g_{27}0000-g_{30}g_{31}00000-g_{33}g_{34}000000-g_{35}-g_{36}000000-g_{36}000000000000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 9
Half sum of positive roots: (4, 7, 9, 10, 10, 9, 7, 4)= \(\displaystyle 4\varepsilon_{1}+3\varepsilon_{2}+2\varepsilon_{3}+\varepsilon_{4}-\varepsilon_{6}-2\varepsilon_{7}-3\varepsilon_{8}-4\varepsilon_{9}\)
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):
(8/9, 7/9, 2/3, 5/9, 4/9, 1/3, 2/9, 1/9) = \(\displaystyle 8/9\varepsilon_{1}-1/9\varepsilon_{2}-1/9\varepsilon_{3}-1/9\varepsilon_{4}-1/9\varepsilon_{5}-1/9\varepsilon_{6}-1/9\varepsilon_{7}-1/9\varepsilon_{8}-1/9\varepsilon_{9}\)
(7/9, 14/9, 4/3, 10/9, 8/9, 2/3, 4/9, 2/9) = \(\displaystyle 7/9\varepsilon_{1}+7/9\varepsilon_{2}-2/9\varepsilon_{3}-2/9\varepsilon_{4}-2/9\varepsilon_{5}-2/9\varepsilon_{6}-2/9\varepsilon_{7}-2/9\varepsilon_{8}-2/9\varepsilon_{9}\)
(2/3, 4/3, 2, 5/3, 4/3, 1, 2/3, 1/3) = \(\displaystyle 2/3\varepsilon_{1}+2/3\varepsilon_{2}+2/3\varepsilon_{3}-1/3\varepsilon_{4}-1/3\varepsilon_{5}-1/3\varepsilon_{6}-1/3\varepsilon_{7}-1/3\varepsilon_{8}-1/3\varepsilon_{9}\)
(5/9, 10/9, 5/3, 20/9, 16/9, 4/3, 8/9, 4/9) = \(\displaystyle 5/9\varepsilon_{1}+5/9\varepsilon_{2}+5/9\varepsilon_{3}+5/9\varepsilon_{4}-4/9\varepsilon_{5}-4/9\varepsilon_{6}-4/9\varepsilon_{7}-4/9\varepsilon_{8}-4/9\varepsilon_{9}\)
(4/9, 8/9, 4/3, 16/9, 20/9, 5/3, 10/9, 5/9) = \(\displaystyle 4/9\varepsilon_{1}+4/9\varepsilon_{2}+4/9\varepsilon_{3}+4/9\varepsilon_{4}+4/9\varepsilon_{5}-5/9\varepsilon_{6}-5/9\varepsilon_{7}-5/9\varepsilon_{8}-5/9\varepsilon_{9}\)
(1/3, 2/3, 1, 4/3, 5/3, 2, 4/3, 2/3) = \(\displaystyle 1/3\varepsilon_{1}+1/3\varepsilon_{2}+1/3\varepsilon_{3}+1/3\varepsilon_{4}+1/3\varepsilon_{5}+1/3\varepsilon_{6}-2/3\varepsilon_{7}-2/3\varepsilon_{8}-2/3\varepsilon_{9}\)
(2/9, 4/9, 2/3, 8/9, 10/9, 4/3, 14/9, 7/9) = \(\displaystyle 2/9\varepsilon_{1}+2/9\varepsilon_{2}+2/9\varepsilon_{3}+2/9\varepsilon_{4}+2/9\varepsilon_{5}+2/9\varepsilon_{6}+2/9\varepsilon_{7}-7/9\varepsilon_{8}-7/9\varepsilon_{9}\)
(1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9) = \(\displaystyle 1/9\varepsilon_{1}+1/9\varepsilon_{2}+1/9\varepsilon_{3}+1/9\varepsilon_{4}+1/9\varepsilon_{5}+1/9\varepsilon_{6}+1/9\varepsilon_{7}+1/9\varepsilon_{8}-8/9\varepsilon_{9}\)
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, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)
(0, 0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{7}-\varepsilon_{8}\)
(0, 0, 0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{8}-\varepsilon_{9}\)