sl(8), type \(A^{1}_7\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: A^{1}_7.
Weyl group size: 40320.
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 56 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-1, -1, -1, -1, -1, -1, -1)-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)-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)-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, -1, -1, -1, -1, -1)-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)-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)-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, -1, -1, -1)-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)-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)-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)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -1, -1)-e_{5}+e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, 0)-e_{4}+e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, -1)-e_{6}+e_{8}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, 0)-e_{5}+e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, -1)-e_{7}+e_{8}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0)-e_{6}+e_{7}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0)e_{6}-e_{7}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1)e_{7}-e_{8}\(s_{7}\)
(1, 1, 0, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0)e_{5}-e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1)e_{6}-e_{8}\(s_{6}s_{7}s_{6}\)
(1, 1, 1, 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)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0)e_{4}-e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1)e_{5}-e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 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)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)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)e_{4}-e_{8}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 1, 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)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)e_{3}-e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 1, 1, 1, 1, 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)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, 1)e_{1}-e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-1, -1, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, 0), (0, 0, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -1, -1, 0, 0), (0, 0, 0, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, -1, -1, -1), (0, 0, 0, -1, -1, -1, 0), (0, 0, -1, -1, -1, 0, 0), (0, -1, -1, -1, 0, 0, 0), (-1, -1, -1, 0, 0, 0, 0), (0, 0, 0, 0, 0, -1, -1), (0, 0, 0, 0, -1, -1, 0), (0, 0, 0, -1, -1, 0, 0), (0, 0, -1, -1, 0, 0, 0), (0, -1, -1, 0, 0, 0, 0), (-1, -1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, -1), (0, 0, 0, 0, 0, -1, 0), (0, 0, 0, 0, -1, 0, 0), (0, 0, 0, -1, 0, 0, 0), (0, 0, -1, 0, 0, 0, 0), (0, -1, 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, 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, 0), (0, 0, 0, 0, 0, 0, 1), (1, 1, 0, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 1, 1), (1, 1, 1, 0, 0, 0, 0), (0, 1, 1, 1, 0, 0, 0), (0, 0, 1, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1, 1), (1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 1, 0), (0, 0, 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) The resulting Lie bracket pairing table follows.
Type A^{1}_7.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_{-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}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}
(-1, -1, -1, -1, -1, -1, -1)-e_{1}+e_{8}g_{-28}0000000000000000000000000000g_{-28}00000g_{-28}g_{-27}00000-g_{-26}g_{-25}0000-g_{-23}g_{-22}000-g_{-19}g_{-18}00-g_{-14}g_{-13}0-g_{-8}g_{-7}-g_{-1}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}
(0, -1, -1, -1, -1, -1, -1)-e_{2}+e_{8}g_{-27}000000000000000000000000000g_{-28}-g_{-27}g_{-27}0000g_{-27}0g_{-25}0000-g_{-24}0g_{-22}000-g_{-20}0g_{-18}00-g_{-15}0g_{-13}0-g_{-9}0g_{-7}-g_{-2}0-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-g_{1}
(-1, -1, -1, -1, -1, -1, 0)-e_{1}+e_{7}g_{-26}000000000000000000000-g_{-28}000000g_{-26}0000g_{-26}-g_{-26}g_{-24}0000-g_{-23}0g_{-21}000-g_{-19}0g_{-17}00-g_{-14}0g_{-12}0-g_{-8}0g_{-6}-g_{-1}0-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}0g_{7}
(0, 0, -1, -1, -1, -1, -1)-e_{3}+e_{8}g_{-25}00000000000000000000g_{-28}00000g_{-27}00-g_{-25}g_{-25}000g_{-25}00g_{-22}000-g_{-21}00g_{-18}00-g_{-16}00g_{-13}0-g_{-10}00g_{-7}-g_{-3}00-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}0-g_{2}-g_{8}
(0, -1, -1, -1, -1, -1, 0)-e_{2}+e_{7}g_{-24}000000000000000000000-g_{-27}00000g_{-26}-g_{-24}g_{-24}000g_{-24}-g_{-24}0g_{-21}000-g_{-20}00g_{-17}00-g_{-15}00g_{-12}0-g_{-9}00g_{-6}-g_{-2}00-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}0-g_{1}g_{7}0
(-1, -1, -1, -1, -1, 0, 0)-e_{1}+e_{6}g_{-23}000000000000000-g_{-28}000000-g_{-26}00000g_{-23}000g_{-23}-g_{-23}0g_{-20}000-g_{-19}00g_{-16}00-g_{-14}00g_{-11}0-g_{-8}00g_{-5}-g_{-1}00-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}00g_{6}0g_{13}
(0, 0, 0, -1, -1, -1, -1)-e_{4}+e_{8}g_{-22}00000000000000g_{-28}0000g_{-27}00000g_{-25}0000-g_{-22}g_{-22}00g_{-22}000g_{-18}00-g_{-17}000g_{-13}0-g_{-11}000g_{-7}-g_{-4}000-h_{7}-h_{6}-h_{5}-h_{4}00-g_{3}0-g_{9}-g_{14}
(0, 0, -1, -1, -1, -1, 0)-e_{3}+e_{7}g_{-21}00000000000000000000g_{-26}-g_{-25}0000g_{-24}00-g_{-21}g_{-21}00g_{-21}-g_{-21}00g_{-17}00-g_{-16}000g_{-12}0-g_{-10}000g_{-6}-g_{-3}000-h_{6}-h_{5}-h_{4}-h_{3}00-g_{2}g_{7}-g_{8}00
(0, -1, -1, -1, -1, 0, 0)-e_{2}+e_{6}g_{-20}000000000000000-g_{-27}000000-g_{-24}0000g_{-23}-g_{-20}g_{-20}00g_{-20}-g_{-20}00g_{-16}00-g_{-15}000g_{-11}0-g_{-9}000g_{-5}-g_{-2}000-h_{5}-h_{4}-h_{3}-h_{2}00-g_{1}g_{6}00g_{13}0
(-1, -1, -1, -1, 0, 0, 0)-e_{1}+e_{5}g_{-19}0000000000-g_{-28}00000-g_{-26}000000-g_{-23}0000g_{-19}00g_{-19}-g_{-19}00g_{-15}00-g_{-14}000g_{-10}0-g_{-8}000g_{-4}-g_{-1}000-h_{4}-h_{3}-h_{2}-h_{1}000g_{5}00g_{12}0g_{18}
(0, 0, 0, 0, -1, -1, -1)-e_{5}+e_{8}g_{-18}000000000g_{-28}000g_{-27}0000g_{-25}00000g_{-22}000000-g_{-18}g_{-18}0g_{-18}0000g_{-13}0-g_{-12}0000g_{-7}-g_{-5}0000-h_{7}-h_{6}-h_{5}000-g_{4}00-g_{10}0-g_{15}-g_{19}
(0, 0, 0, -1, -1, -1, 0)-e_{4}+e_{7}g_{-17}00000000000000g_{-26}0000g_{-24}0-g_{-22}000g_{-21}0000-g_{-17}g_{-17}0g_{-17}-g_{-17}000g_{-12}0-g_{-11}0000g_{-6}-g_{-4}0000-h_{6}-h_{5}-h_{4}000-g_{3}g_{7}0-g_{9}0-g_{14}00
(0, 0, -1, -1, -1, 0, 0)-e_{3}+e_{6}g_{-16}000000000000000-g_{-25}0000g_{-23}0-g_{-21}000g_{-20}00-g_{-16}g_{-16}0g_{-16}-g_{-16}000g_{-11}0-g_{-10}0000g_{-5}-g_{-3}0000-h_{5}-h_{4}-h_{3}000-g_{2}g_{6}0-g_{8}0g_{13}000
(0, -1, -1, -1, 0, 0, 0)-e_{2}+e_{5}g_{-15}0000000000-g_{-27}00000-g_{-24}000000-g_{-20}000g_{-19}-g_{-15}g_{-15}0g_{-15}-g_{-15}000g_{-10}0-g_{-9}0000g_{-4}-g_{-2}0000-h_{4}-h_{3}-h_{2}000-g_{1}g_{5}000g_{12}00g_{18}0
(-1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{4}g_{-14}000000-g_{-28}0000-g_{-26}00000-g_{-23}000000-g_{-19}000g_{-14}0g_{-14}-g_{-14}000g_{-9}0-g_{-8}0000g_{-3}-g_{-1}0000-h_{3}-h_{2}-h_{1}0000g_{4}000g_{11}00g_{17}0g_{22}
(0, 0, 0, 0, 0, -1, -1)-e_{6}+e_{8}g_{-13}00000g_{-28}00g_{-27}000g_{-25}0000g_{-22}00000g_{-18}00000000-g_{-13}g_{-13}g_{-13}00000g_{-7}-g_{-6}00000-h_{7}-h_{6}0000-g_{5}000-g_{11}00-g_{16}0-g_{20}-g_{23}
(0, 0, 0, 0, -1, -1, 0)-e_{5}+e_{7}g_{-12}000000000g_{-26}000g_{-24}0000g_{-21}00-g_{-18}00g_{-17}000000-g_{-12}g_{-12}g_{-12}-g_{-12}0000g_{-6}-g_{-5}00000-h_{6}-h_{5}0000-g_{4}g_{7}00-g_{10}00-g_{15}0-g_{19}00
(0, 0, 0, -1, -1, 0, 0)-e_{4}+e_{6}g_{-11}00000000000000g_{-23}-g_{-22}000g_{-20}00-g_{-17}00g_{-16}0000-g_{-11}g_{-11}g_{-11}-g_{-11}0000g_{-5}-g_{-4}00000-h_{5}-h_{4}0000-g_{3}g_{6}00-g_{9}0g_{13}-g_{14}00000
(0, 0, -1, -1, 0, 0, 0)-e_{3}+e_{5}g_{-10}0000000000-g_{-25}00000-g_{-21}000g_{-19}00-g_{-16}00g_{-15}00-g_{-10}g_{-10}g_{-10}-g_{-10}0000g_{-4}-g_{-3}00000-h_{4}-h_{3}0000-g_{2}g_{5}00-g_{8}0g_{12}000g_{18}000
(0, -1, -1, 0, 0, 0, 0)-e_{2}+e_{4}g_{-9}000000-g_{-27}0000-g_{-24}00000-g_{-20}000000-g_{-15}00g_{-14}-g_{-9}g_{-9}g_{-9}-g_{-9}0000g_{-3}-g_{-2}00000-h_{3}-h_{2}0000-g_{1}g_{4}0000g_{11}000g_{17}00g_{22}0
(-1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{3}g_{-8}000-g_{-28}000-g_{-26}0000-g_{-23}00000-g_{-19}000000-g_{-14}00g_{-8}g_{-8}-g_{-8}0000g_{-2}-g_{-1}00000-h_{2}-h_{1}00000g_{3}0000g_{10}000g_{16}00g_{21}0g_{25}
(0, 0, 0, 0, 0, 0, -1)-e_{7}+e_{8}g_{-7}00g_{-28}0g_{-27}00g_{-25}000g_{-22}0000g_{-18}00000g_{-13}0000000000-g_{-7}2g_{-7}000000-h_{7}00000-g_{6}0000-g_{12}000-g_{17}00-g_{21}0-g_{24}-g_{26}
(0, 0, 0, 0, 0, -1, 0)-e_{6}+e_{7}g_{-6}00000g_{-26}00g_{-24}000g_{-21}0000g_{-17}000-g_{-13}0g_{-12}00000000-g_{-6}2g_{-6}-g_{-6}00000-h_{6}00000-g_{5}g_{7}000-g_{11}000-g_{16}00-g_{20}0-g_{23}00
(0, 0, 0, 0, -1, 0, 0)-e_{5}+e_{6}g_{-5}000000000g_{-23}000g_{-20}0-g_{-18}00g_{-16}000-g_{-12}0g_{-11}000000-g_{-5}2g_{-5}-g_{-5}00000-h_{5}00000-g_{4}g_{6}000-g_{10}0g_{13}0-g_{15}00-g_{19}00000
(0, 0, 0, -1, 0, 0, 0)-e_{4}+e_{5}g_{-4}0000000000-g_{-22}000g_{-19}0-g_{-17}00g_{-15}000-g_{-11}0g_{-10}0000-g_{-4}2g_{-4}-g_{-4}00000-h_{4}00000-g_{3}g_{5}000-g_{9}0g_{12}0-g_{14}00g_{18}000000
(0, 0, -1, 0, 0, 0, 0)-e_{3}+e_{4}g_{-3}000000-g_{-25}0000-g_{-21}00000-g_{-16}00g_{-14}000-g_{-10}0g_{-9}00-g_{-3}2g_{-3}-g_{-3}00000-h_{3}00000-g_{2}g_{4}000-g_{8}0g_{11}0000g_{17}000g_{22}000
(0, -1, 0, 0, 0, 0, 0)-e_{2}+e_{3}g_{-2}000-g_{-27}000-g_{-24}0000-g_{-20}00000-g_{-15}000000-g_{-9}0g_{-8}-g_{-2}2g_{-2}-g_{-2}00000-h_{2}00000-g_{1}g_{3}00000g_{10}0000g_{16}000g_{21}00g_{25}0
(-1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}g_{-1}0-g_{-28}00-g_{-26}000-g_{-23}0000-g_{-19}00000-g_{-14}000000-g_{-8}02g_{-1}-g_{-1}00000-h_{1}000000g_{2}00000g_{9}0000g_{15}000g_{20}00g_{24}0g_{27}
(0, 0, 0, 0, 0, 0, 0)0h_{1}-g_{-28}g_{-27}-g_{-26}0g_{-24}-g_{-23}00g_{-20}-g_{-19}000g_{-15}-g_{-14}0000g_{-9}-g_{-8}00000g_{-2}-2g_{-1}00000002g_{1}-g_{2}00000g_{8}-g_{9}0000g_{14}-g_{15}000g_{19}-g_{20}00g_{23}-g_{24}0g_{26}-g_{27}g_{28}
(0, 0, 0, 0, 0, 0, 0)0h_{2}0-g_{-27}0g_{-25}-g_{-24}00g_{-21}-g_{-20}000g_{-16}-g_{-15}0000g_{-10}-g_{-9}-g_{-8}0000g_{-3}-2g_{-2}g_{-1}0000000-g_{1}2g_{2}-g_{3}0000g_{8}g_{9}-g_{10}0000g_{15}-g_{16}000g_{20}-g_{21}00g_{24}-g_{25}0g_{27}0
(0, 0, 0, 0, 0, 0, 0)0h_{3}000-g_{-25}00g_{-22}-g_{-21}000g_{-17}-g_{-16}0-g_{-14}00g_{-11}-g_{-10}-g_{-9}g_{-8}000g_{-4}-2g_{-3}g_{-2}000000000-g_{2}2g_{3}-g_{4}000-g_{8}g_{9}g_{10}-g_{11}00g_{14}0g_{16}-g_{17}000g_{21}-g_{22}00g_{25}000
(0, 0, 0, 0, 0, 0, 0)0h_{4}000000-g_{-22}00-g_{-19}g_{-18}-g_{-17}0-g_{-15}g_{-14}0g_{-12}-g_{-11}-g_{-10}g_{-9}000g_{-5}-2g_{-4}g_{-3}00000000000-g_{3}2g_{4}-g_{5}000-g_{9}g_{10}g_{11}-g_{12}0-g_{14}g_{15}0g_{17}-g_{18}g_{19}00g_{22}000000
(0, 0, 0, 0, 0, 0, 0)0h_{5}00000-g_{-23}00-g_{-20}g_{-19}-g_{-18}0-g_{-16}g_{-15}0g_{-13}-g_{-12}-g_{-11}g_{-10}000g_{-6}-2g_{-5}g_{-4}0000000000000-g_{4}2g_{5}-g_{6}000-g_{10}g_{11}g_{12}-g_{13}0-g_{15}g_{16}0g_{18}-g_{19}g_{20}00g_{23}00000
(0, 0, 0, 0, 0, 0, 0)0h_{6}00-g_{-26}0-g_{-24}g_{-23}0-g_{-21}g_{-20}00-g_{-17}g_{-16}00-g_{-13}-g_{-12}g_{-11}000g_{-7}-2g_{-6}g_{-5}000000000000000-g_{5}2g_{6}-g_{7}000-g_{11}g_{12}g_{13}00-g_{16}g_{17}00-g_{20}g_{21}0-g_{23}g_{24}0g_{26}00
(0, 0, 0, 0, 0, 0, 0)0h_{7}-g_{-28}-g_{-27}g_{-26}-g_{-25}g_{-24}0-g_{-22}g_{-21}00-g_{-18}g_{-17}000-g_{-13}g_{-12}0000-2g_{-7}g_{-6}00000000000000000-g_{6}2g_{7}0000-g_{12}g_{13}000-g_{17}g_{18}00-g_{21}g_{22}0-g_{24}g_{25}-g_{26}g_{27}g_{28}
(1, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}g_{1}-g_{-27}0-g_{-24}00-g_{-20}000-g_{-15}0000-g_{-9}00000-g_{-2}000000h_{1}-2g_{1}g_{1}000000g_{8}000000g_{14}00000g_{19}0000g_{23}000g_{26}00g_{28}0
(0, 1, 0, 0, 0, 0, 0)e_{2}-e_{3}g_{2}0-g_{-25}00-g_{-21}000-g_{-16}0000-g_{-10}00000-g_{-3}g_{-1}00000h_{2}0g_{2}-2g_{2}g_{2}0000-g_{8}0g_{9}000000g_{15}00000g_{20}0000g_{24}000g_{27}000
(0, 0, 1, 0, 0, 0, 0)e_{3}-e_{4}g_{3}000-g_{-22}000-g_{-17}0000-g_{-11}0g_{-8}000-g_{-4}g_{-2}00000h_{3}000g_{3}-2g_{3}g_{3}0000-g_{9}0g_{10}000-g_{14}00g_{16}00000g_{21}0000g_{25}000000
(0, 0, 0, 1, 0, 0, 0)e_{4}-e_{5}g_{4}000000-g_{-18}00g_{-14}0-g_{-12}0g_{-9}000-g_{-5}g_{-3}00000h_{4}00000g_{4}-2g_{4}g_{4}0000-g_{10}0g_{11}000-g_{15}00g_{17}0-g_{19}000g_{22}0000000000
(0, 0, 0, 0, 1, 0, 0)e_{5}-e_{6}g_{5}00000g_{-19}00g_{-15}0-g_{-13}0g_{-10}000-g_{-6}g_{-4}00000h_{5}0000000g_{5}-2g_{5}g_{5}0000-g_{11}0g_{12}000-g_{16}00g_{18}0-g_{20}000-g_{23}000000000
(0, 0, 0, 0, 0, 1, 0)e_{6}-e_{7}g_{6}00g_{-23}0g_{-20}00g_{-16}000g_{-11}000-g_{-7}g_{-5}00000h_{6}000000000g_{6}-2g_{6}g_{6}0000-g_{12}0g_{13}000-g_{17}0000-g_{21}000-g_{24}00-g_{26}00000
(0, 0, 0, 0, 0, 0, 1)e_{7}-e_{8}g_{7}g_{-26}g_{-24}0g_{-21}00g_{-17}000g_{-12}0000g_{-6}00000h_{7}00000000000g_{7}-2g_{7}00000-g_{13}00000-g_{18}0000-g_{22}000-g_{25}00-g_{27}0-g_{28}00
(1, 1, 0, 0, 0, 0, 0)e_{1}-e_{3}g_{8}-g_{-25}0-g_{-21}00-g_{-16}000-g_{-10}0000-g_{-3}00000h_{2}+h_{1}00000g_{1}-g_{2}-g_{8}-g_{8}g_{8}000000g_{14}000000g_{19}00000g_{23}0000g_{26}000g_{28}000
(0, 1, 1, 0, 0, 0, 0)e_{2}-e_{4}g_{9}0-g_{-22}00-g_{-17}000-g_{-11}0000-g_{-4}g_{-1}0000h_{3}+h_{2}00000g_{2}-g_{3}0g_{9}-g_{9}-g_{9}g_{9}000-g_{14}00g_{15}000000g_{20}00000g_{24}0000g_{27}000000
(0, 0, 1, 1, 0, 0, 0)e_{3}-e_{5}g_{10}000-g_{-18}000-g_{-12}0g_{-8}00-g_{-5}g_{-2}0000h_{4}+h_{3}00000g_{3}-g_{4}000g_{10}-g_{10}-g_{10}g_{10}000-g_{15}00g_{16}00-g_{19}000g_{21}00000g_{25}0000000000
(0, 0, 0, 1, 1, 0, 0)e_{4}-e_{6}g_{11}00000g_{-14}-g_{-13}0g_{-9}00-g_{-6}g_{-3}0000h_{5}+h_{4}00000g_{4}-g_{5}00000g_{11}-g_{11}-g_{11}g_{11}000-g_{16}00g_{17}00-g_{20}000g_{22}-g_{23}00000000000000
(0, 0, 0, 0, 1, 1, 0)e_{5}-e_{7}g_{12}00g_{-19}0g_{-15}00g_{-10}00-g_{-7}g_{-4}0000h_{6}+h_{5}00000g_{5}-g_{6}0000000g_{12}-g_{12}-g_{12}g_{12}000-g_{17}00g_{18}00-g_{21}0000-g_{24}000-g_{26}000000000
(0, 0, 0, 0, 0, 1, 1)e_{6}-e_{8}g_{13}g_{-23}g_{-20}0g_{-16}00g_{-11}000g_{-5}0000h_{7}+h_{6}00000g_{6}-g_{7}000000000g_{13}-g_{13}-g_{13}0000-g_{18}00000-g_{22}0000-g_{25}000-g_{27}00-g_{28}00000
(1, 1, 1, 0, 0, 0, 0)e_{1}-e_{4}g_{14}-g_{-22}0-g_{-17}00-g_{-11}000-g_{-4}0000h_{3}+h_{2}+h_{1}0000g_{1}-g_{3}0000g_{8}0-g_{9}-g_{14}0-g_{14}g_{14}000000g_{19}000000g_{23}00000g_{26}0000g_{28}000000
(0, 1, 1, 1, 0, 0, 0)e_{2}-e_{5}g_{15}0-g_{-18}00-g_{-12}000-g_{-5}g_{-1}000h_{4}+h_{3}+h_{2}0000g_{2}-g_{4}0000g_{9}0-g_{10}0g_{15}-g_{15}0-g_{15}g_{15}00-g_{19}000g_{20}000000g_{24}00000g_{27}0000000000
(0, 0, 1, 1, 1, 0, 0)e_{3}-e_{6}g_{16}000-g_{-13}0g_{-8}0-g_{-6}g_{-2}000h_{5}+h_{4}+h_{3}0000g_{3}-g_{5}0000g_{10}0-g_{11}000g_{16}-g_{16}0-g_{16}g_{16}00-g_{20}000g_{21}0-g_{23}0000g_{25}000000000000000
(0, 0, 0, 1, 1, 1, 0)e_{4}-e_{7}g_{17}00g_{-14}0g_{-9}0-g_{-7}g_{-3}000h_{6}+h_{5}+h_{4}0000g_{4}-g_{6}0000g_{11}0-g_{12}00000g_{17}-g_{17}0-g_{17}g_{17}00-g_{21}000g_{22}0-g_{24}0000-g_{26}00000000000000
(0, 0, 0, 0, 1, 1, 1)e_{5}-e_{8}g_{18}g_{-19}g_{-15}0g_{-10}00g_{-4}000h_{7}+h_{6}+h_{5}0000g_{5}-g_{7}0000g_{12}0-g_{13}0000000g_{18}-g_{18}0-g_{18}000-g_{22}00000-g_{25}0000-g_{27}000-g_{28}000000000
(1, 1, 1, 1, 0, 0, 0)e_{1}-e_{5}g_{19}-g_{-18}0-g_{-12}00-g_{-5}000h_{4}+h_{3}+h_{2}+h_{1}000g_{1}-g_{4}000g_{8}0-g_{10}000g_{14}00-g_{15}-g_{19}00-g_{19}g_{19}000000g_{23}000000g_{26}00000g_{28}0000000000
(0, 1, 1, 1, 1, 0, 0)e_{2}-e_{6}g_{20}0-g_{-13}00-g_{-6}g_{-1}00h_{5}+h_{4}+h_{3}+h_{2}000g_{2}-g_{5}000g_{9}0-g_{11}000g_{15}00-g_{16}0g_{20}-g_{20}00-g_{20}g_{20}0-g_{23}0000g_{24}000000g_{27}000000000000000
(0, 0, 1, 1, 1, 1, 0)e_{3}-e_{7}g_{21}00g_{-8}-g_{-7}g_{-2}00h_{6}+h_{5}+h_{4}+h_{3}000g_{3}-g_{6}000g_{10}0-g_{12}000g_{16}00-g_{17}000g_{21}-g_{21}00-g_{21}g_{21}0-g_{24}0000g_{25}-g_{26}00000000000000000000
(0, 0, 0, 1, 1, 1, 1)e_{4}-e_{8}g_{22}g_{-14}g_{-9}0g_{-3}00h_{7}+h_{6}+h_{5}+h_{4}000g_{4}-g_{7}000g_{11}0-g_{13}000g_{17}00-g_{18}00000g_{22}-g_{22}00-g_{22}00-g_{25}00000-g_{27}0000-g_{28}00000000000000
(1, 1, 1, 1, 1, 0, 0)e_{1}-e_{6}g_{23}-g_{-13}0-g_{-6}00h_{5}+h_{4}+h_{3}+h_{2}+h_{1}00g_{1}-g_{5}00g_{8}0-g_{11}00g_{14}00-g_{16}00g_{19}000-g_{20}-g_{23}000-g_{23}g_{23}000000g_{26}000000g_{28}000000000000000
(0, 1, 1, 1, 1, 1, 0)e_{2}-e_{7}g_{24}0-g_{-7}g_{-1}0h_{6}+h_{5}+h_{4}+h_{3}+h_{2}00g_{2}-g_{6}00g_{9}0-g_{12}00g_{15}00-g_{17}00g_{20}000-g_{21}0g_{24}-g_{24}000-g_{24}g_{24}-g_{26}00000g_{27}000000000000000000000
(0, 0, 1, 1, 1, 1, 1)e_{3}-e_{8}g_{25}g_{-8}g_{-2}0h_{7}+h_{6}+h_{5}+h_{4}+h_{3}00g_{3}-g_{7}00g_{10}0-g_{13}00g_{16}00-g_{18}00g_{21}000-g_{22}000g_{25}-g_{25}000-g_{25}0-g_{27}00000-g_{28}00000000000000000000
(1, 1, 1, 1, 1, 1, 0)e_{1}-e_{7}g_{26}-g_{-7}0h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}-g_{6}0g_{8}0-g_{12}0g_{14}00-g_{17}0g_{19}000-g_{21}0g_{23}0000-g_{24}-g_{26}0000-g_{26}g_{26}000000g_{28}000000000000000000000
(0, 1, 1, 1, 1, 1, 1)e_{2}-e_{8}g_{27}g_{-1}h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}0g_{2}-g_{7}0g_{9}0-g_{13}0g_{15}00-g_{18}0g_{20}000-g_{22}0g_{24}0000-g_{25}0g_{27}-g_{27}0000-g_{27}-g_{28}000000000000000000000000000
(1, 1, 1, 1, 1, 1, 1)e_{1}-e_{8}g_{28}h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}g_{1}-g_{7}g_{8}0-g_{13}g_{14}00-g_{18}g_{19}000-g_{22}g_{23}0000-g_{25}g_{26}00000-g_{27}-g_{28}00000-g_{28}0000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 8
Half sum of positive roots: (7/2, 6, 15/2, 8, 15/2, 6, 7/2)= \(\displaystyle 7/2\varepsilon_{1}+5/2\varepsilon_{2}+3/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-3/2\varepsilon_{6}-5/2\varepsilon_{7}-7/2\varepsilon_{8}\)
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):
(7/8, 3/4, 5/8, 1/2, 3/8, 1/4, 1/8) = \(\displaystyle 7/8\varepsilon_{1}-1/8\varepsilon_{2}-1/8\varepsilon_{3}-1/8\varepsilon_{4}-1/8\varepsilon_{5}-1/8\varepsilon_{6}-1/8\varepsilon_{7}-1/8\varepsilon_{8}\)
(3/4, 3/2, 5/4, 1, 3/4, 1/2, 1/4) = \(\displaystyle 3/4\varepsilon_{1}+3/4\varepsilon_{2}-1/4\varepsilon_{3}-1/4\varepsilon_{4}-1/4\varepsilon_{5}-1/4\varepsilon_{6}-1/4\varepsilon_{7}-1/4\varepsilon_{8}\)
(5/8, 5/4, 15/8, 3/2, 9/8, 3/4, 3/8) = \(\displaystyle 5/8\varepsilon_{1}+5/8\varepsilon_{2}+5/8\varepsilon_{3}-3/8\varepsilon_{4}-3/8\varepsilon_{5}-3/8\varepsilon_{6}-3/8\varepsilon_{7}-3/8\varepsilon_{8}\)
(1/2, 1, 3/2, 2, 3/2, 1, 1/2) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}-1/2\varepsilon_{8}\)
(3/8, 3/4, 9/8, 3/2, 15/8, 5/4, 5/8) = \(\displaystyle 3/8\varepsilon_{1}+3/8\varepsilon_{2}+3/8\varepsilon_{3}+3/8\varepsilon_{4}+3/8\varepsilon_{5}-5/8\varepsilon_{6}-5/8\varepsilon_{7}-5/8\varepsilon_{8}\)
(1/4, 1/2, 3/4, 1, 5/4, 3/2, 3/4) = \(\displaystyle 1/4\varepsilon_{1}+1/4\varepsilon_{2}+1/4\varepsilon_{3}+1/4\varepsilon_{4}+1/4\varepsilon_{5}+1/4\varepsilon_{6}-3/4\varepsilon_{7}-3/4\varepsilon_{8}\)
(1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8) = \(\displaystyle 1/8\varepsilon_{1}+1/8\varepsilon_{2}+1/8\varepsilon_{3}+1/8\varepsilon_{4}+1/8\varepsilon_{5}+1/8\varepsilon_{6}+1/8\varepsilon_{7}-7/8\varepsilon_{8}\)
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) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)
(0, 0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{7}-\varepsilon_{8}\)