sp(10), type \(C^{1}_5\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.
Page generated by the calculator project.
Lie algebra type: C^{1}_5.
Weyl group size: 3840.
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 50 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-2, -2, -2, -2, -1)-2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(-1, -2, -2, -2, -1)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{3}s_{2}\)
(0, -2, -2, -2, -1)-2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -1)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{3}\)
(0, -1, -2, -2, -1)-e_{2}-e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{5}s_{4}s_{3}\)
(-1, -1, -1, -2, -1)-e_{1}-e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}\)
(0, 0, -2, -2, -1)-2e_{3}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -2, -1)-e_{2}-e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{5}s_{4}\)
(-1, -1, -1, -1, -1)-e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}\)
(0, 0, -1, -2, -1)-e_{3}-e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{3}s_{5}s_{4}\)
(0, -1, -1, -1, -1)-e_{2}-e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{3}s_{2}s_{5}\)
(-1, -1, -1, -1, 0)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, -2, -1)-2e_{4}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, -1)-e_{3}-e_{5}\(s_{3}s_{5}s_{4}s_{3}s_{5}\)
(0, -1, -1, -1, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, -1, -1)-e_{4}-e_{5}\(s_{5}s_{4}s_{5}\)
(0, 0, -1, -1, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, -1)-2e_{5}\(s_{5}\)
(0, 0, 0, -1, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1)2e_{5}\(s_{5}\)
(1, 1, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1)e_{4}+e_{5}\(s_{5}s_{4}s_{5}\)
(1, 1, 1, 0, 0)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 0)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{3}s_{5}\)
(0, 0, 0, 2, 1)2e_{4}\(s_{4}s_{5}s_{4}\)
(1, 1, 1, 1, 0)e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1)e_{2}+e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{3}s_{2}s_{5}\)
(0, 0, 1, 2, 1)e_{3}+e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{3}s_{5}s_{4}\)
(1, 1, 1, 1, 1)e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}\)
(0, 1, 1, 2, 1)e_{2}+e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{5}s_{4}\)
(0, 0, 2, 2, 1)2e_{3}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(1, 1, 1, 2, 1)e_{1}+e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}\)
(0, 1, 2, 2, 1)e_{2}+e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 1)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{3}\)
(0, 2, 2, 2, 1)2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(1, 2, 2, 2, 1)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{3}s_{2}\)
(2, 2, 2, 2, 1)2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-2, -2, -2, -2, -1), (-1, -2, -2, -2, -1), (0, -2, -2, -2, -1), (-1, -1, -2, -2, -1), (0, -1, -2, -2, -1), (-1, -1, -1, -2, -1), (0, 0, -2, -2, -1), (0, -1, -1, -2, -1), (-1, -1, -1, -1, -1), (0, 0, -1, -2, -1), (0, -1, -1, -1, -1), (-1, -1, -1, -1, 0), (0, 0, 0, -2, -1), (0, 0, -1, -1, -1), (0, -1, -1, -1, 0), (-1, -1, -1, 0, 0), (0, 0, 0, -1, -1), (0, 0, -1, -1, 0), (0, -1, -1, 0, 0), (-1, -1, 0, 0, 0), (0, 0, 0, 0, -1), (0, 0, 0, -1, 0), (0, 0, -1, 0, 0), (0, -1, 0, 0, 0), (-1, 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), (1, 1, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 1, 1, 0), (0, 0, 0, 1, 1), (1, 1, 1, 0, 0), (0, 1, 1, 1, 0), (0, 0, 1, 1, 1), (0, 0, 0, 2, 1), (1, 1, 1, 1, 0), (0, 1, 1, 1, 1), (0, 0, 1, 2, 1), (1, 1, 1, 1, 1), (0, 1, 1, 2, 1), (0, 0, 2, 2, 1), (1, 1, 1, 2, 1), (0, 1, 2, 2, 1), (1, 1, 2, 2, 1), (0, 2, 2, 2, 1), (1, 2, 2, 2, 1), (2, 2, 2, 2, 1) The resulting Lie bracket pairing table follows.
Type C^{1}_5.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_{-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}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}
(-2, -2, -2, -2, -1)-2e_{1}g_{-25}0000000000000000000000000g_{-25}0000g_{-24}0000g_{-22}000g_{-20}000g_{-17}00-g_{-14}00-g_{-10}0-g_{-6}0-g_{-1}-h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}
(-1, -2, -2, -2, -1)-e_{1}-e_{2}g_{-24}0000000000000000000000002g_{-25}01/2g_{-24}0002g_{-23}g_{-22}000g_{-21}g_{-20}00g_{-18}g_{-17}00g_{-15}-g_{-14}0-g_{-11}-g_{-10}0-g_{-7}-g_{-6}-g_{-2}-g_{-1}-2h_{5}-4h_{4}-4h_{3}-4h_{2}-2h_{1}-g_{1}
(0, -2, -2, -2, -1)-2e_{2}g_{-23}000000000000000000000000g_{-24}-g_{-23}g_{-23}0000g_{-21}0000g_{-18}000g_{-15}000-g_{-11}00-g_{-7}00-g_{-2}0-h_{5}-2h_{4}-2h_{3}-2h_{2}-g_{1}0
(-1, -1, -2, -2, -1)-e_{1}-e_{3}g_{-22}00000000000000000002g_{-25}000g_{-24}01/2g_{-22}-1/2g_{-22}1/2g_{-22}00g_{-21}0g_{-20}002g_{-19}0g_{-17}0g_{-16}0-g_{-14}0g_{-12}0-g_{-10}-g_{-8}0-g_{-6}-g_{-3}-g_{-1}-2h_{5}-4h_{4}-4h_{3}-2h_{2}-2h_{1}0-g_{2}-g_{6}
(0, -1, -2, -2, -1)-e_{2}-e_{3}g_{-21}0000000000000000000g_{-24}0002g_{-23}g_{-22}-1/2g_{-21}01/2g_{-21}0002g_{-19}g_{-18}000g_{-16}g_{-15}00g_{-12}-g_{-11}00-g_{-8}-g_{-7}0-g_{-3}-g_{-2}0-2h_{5}-4h_{4}-4h_{3}-2h_{2}-g_{1}-g_{2}-g_{6}0
(-1, -1, -1, -2, -1)-e_{1}-e_{4}g_{-20}0000000000000002g_{-25}00g_{-24}000g_{-22}001/2g_{-20}0-1/2g_{-20}1/2g_{-20}0g_{-18}00g_{-17}0g_{-16}00-g_{-14}2g_{-13}00-g_{-10}g_{-9}0-g_{-6}-g_{-4}-g_{-1}0-2h_{5}-4h_{4}-2h_{3}-2h_{2}-2h_{1}0-g_{3}0-g_{7}-g_{10}
(0, 0, -2, -2, -1)-2e_{3}g_{-19}0000000000000000000g_{-22}000g_{-21}00-g_{-19}g_{-19}0000g_{-16}0000g_{-12}000-g_{-8}000-g_{-3}00-h_{5}-2h_{4}-2h_{3}0-g_{2}-g_{6}000
(0, -1, -1, -2, -1)-e_{2}-e_{4}g_{-18}000000000000000g_{-24}002g_{-23}000g_{-21}0g_{-20}-1/2g_{-18}1/2g_{-18}-1/2g_{-18}1/2g_{-18}00g_{-16}0g_{-15}002g_{-13}0-g_{-11}0g_{-9}0-g_{-7}0-g_{-4}-g_{-2}0-2h_{5}-4h_{4}-2h_{3}-2h_{2}0-g_{1}-g_{3}0-g_{7}-g_{10}0
(-1, -1, -1, -1, -1)-e_{1}-e_{5}g_{-17}000000000002g_{-25}00g_{-24}00g_{-22}000g_{-20}0001/2g_{-17}00-1/2g_{-17}g_{-17}g_{-15}000-g_{-14}g_{-12}00-g_{-10}g_{-9}0-g_{-6}02g_{-5}-g_{-1}0-2h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}00-g_{4}0-g_{8}0-g_{11}-g_{14}
(0, 0, -1, -2, -1)-e_{3}-e_{4}g_{-16}000000000000000g_{-22}00g_{-21}g_{-20}002g_{-19}g_{-18}00-1/2g_{-16}01/2g_{-16}0002g_{-13}g_{-12}000g_{-9}-g_{-8}00-g_{-4}-g_{-3}00-2h_{5}-4h_{4}-2h_{3}0-g_{2}-g_{3}-g_{6}-g_{7}-g_{10}000
(0, -1, -1, -1, -1)-e_{2}-e_{5}g_{-15}00000000000g_{-24}002g_{-23}00g_{-21}000g_{-18}00g_{-17}-1/2g_{-15}1/2g_{-15}0-1/2g_{-15}g_{-15}0g_{-12}00-g_{-11}0g_{-9}0-g_{-7}02g_{-5}-g_{-2}00-2h_{5}-2h_{4}-2h_{3}-2h_{2}0-g_{1}-g_{4}00-g_{8}0-g_{11}-g_{14}0
(-1, -1, -1, -1, 0)-e_{1}+e_{5}g_{-14}00000000-2g_{-25}0-g_{-24}00-g_{-22}00-g_{-20}000-g_{-17}00001/2g_{-14}001/2g_{-14}-g_{-14}g_{-11}00-g_{-10}0g_{-8}0-g_{-6}0g_{-4}-g_{-1}00-2h_{4}-2h_{3}-2h_{2}-2h_{1}002g_{5}00g_{9}0g_{12}0g_{15}g_{17}
(0, 0, 0, -2, -1)-2e_{4}g_{-13}000000000000000g_{-20}00g_{-18}000g_{-16}0000-g_{-13}g_{-13}0000g_{-9}0000-g_{-4}000-h_{5}-2h_{4}00-g_{3}0-g_{7}0-g_{10}00000
(0, 0, -1, -1, -1)-e_{3}-e_{5}g_{-12}00000000000g_{-22}00g_{-21}002g_{-19}0g_{-17}0g_{-16}0g_{-15}00-1/2g_{-12}1/2g_{-12}-1/2g_{-12}g_{-12}00g_{-9}0-g_{-8}002g_{-5}-g_{-3}00-2h_{5}-2h_{4}-2h_{3}00-g_{2}-g_{4}-g_{6}0-g_{8}0-g_{11}-g_{14}000
(0, -1, -1, -1, 0)-e_{2}+e_{5}g_{-11}00000000-g_{-24}0-2g_{-23}00-g_{-21}00-g_{-18}000-g_{-15}000g_{-14}-1/2g_{-11}1/2g_{-11}01/2g_{-11}-g_{-11}0g_{-8}0-g_{-7}00g_{-4}-g_{-2}00-2h_{4}-2h_{3}-2h_{2}00-g_{1}2g_{5}00g_{9}00g_{12}0g_{15}g_{17}0
(-1, -1, -1, 0, 0)-e_{1}+e_{4}g_{-10}00000-2g_{-25}0-g_{-24}0-g_{-22}00-g_{-20}000-g_{-17}0000-g_{-14}0001/2g_{-10}01/2g_{-10}-1/2g_{-10}0g_{-7}0-g_{-6}00g_{-3}-g_{-1}00-2h_{3}-2h_{2}-2h_{1}000g_{4}00g_{9}002g_{13}0g_{16}0g_{18}g_{20}
(0, 0, 0, -1, -1)-e_{4}-e_{5}g_{-9}00000000000g_{-20}00g_{-18}g_{-17}0g_{-16}g_{-15}002g_{-13}g_{-12}0000-1/2g_{-9}0g_{-9}0002g_{-5}-g_{-4}000-2h_{5}-2h_{4}00-g_{3}-g_{4}0-g_{7}-g_{8}-g_{10}-g_{11}0-g_{14}00000
(0, 0, -1, -1, 0)-e_{3}+e_{5}g_{-8}00000000-g_{-22}0-g_{-21}00-2g_{-19}00-g_{-16}00g_{-14}-g_{-12}00g_{-11}00-1/2g_{-8}1/2g_{-8}1/2g_{-8}-g_{-8}00g_{-4}-g_{-3}000-2h_{4}-2h_{3}00-g_{2}2g_{5}0-g_{6}0g_{9}00g_{12}0g_{15}g_{17}000
(0, -1, -1, 0, 0)-e_{2}+e_{4}g_{-7}00000-g_{-24}0-2g_{-23}0-g_{-21}00-g_{-18}000-g_{-15}0000-g_{-11}00g_{-10}-1/2g_{-7}1/2g_{-7}1/2g_{-7}-1/2g_{-7}00g_{-3}-g_{-2}000-2h_{3}-2h_{2}00-g_{1}g_{4}000g_{9}002g_{13}00g_{16}0g_{18}g_{20}0
(-1, -1, 0, 0, 0)-e_{1}+e_{3}g_{-6}000-2g_{-25}-g_{-24}0-g_{-22}00-g_{-20}000-g_{-17}000-g_{-14}0000-g_{-10}001/2g_{-6}1/2g_{-6}-1/2g_{-6}00g_{-2}-g_{-1}000-2h_{2}-2h_{1}000g_{3}000g_{8}00g_{12}00g_{16}02g_{19}0g_{21}g_{22}
(0, 0, 0, 0, -1)-2e_{5}g_{-5}00000000000g_{-17}00g_{-15}00g_{-12}000g_{-9}000000-g_{-5}2g_{-5}0000-h_{5}000-g_{4}00-g_{8}00-g_{11}0-g_{14}00000000
(0, 0, 0, -1, 0)-e_{4}+e_{5}g_{-4}00000000-g_{-20}0-g_{-18}00-g_{-16}0g_{-14}-2g_{-13}0g_{-11}0-g_{-9}0g_{-8}0000-1/2g_{-4}g_{-4}-g_{-4}000-2h_{4}000-g_{3}2g_{5}0-g_{7}0g_{9}-g_{10}0g_{12}0g_{15}0g_{17}00000
(0, 0, -1, 0, 0)-e_{3}+e_{4}g_{-3}00000-g_{-22}0-g_{-21}0-2g_{-19}00-g_{-16}000-g_{-12}00g_{-10}0-g_{-8}0g_{-7}00-1/2g_{-3}g_{-3}-1/2g_{-3}000-2h_{3}000-g_{2}g_{4}0-g_{6}0g_{9}0002g_{13}00g_{16}0g_{18}g_{20}000
(0, -1, 0, 0, 0)-e_{2}+e_{3}g_{-2}000-g_{-24}-2g_{-23}0-g_{-21}00-g_{-18}000-g_{-15}000-g_{-11}0000-g_{-7}0g_{-6}-1/2g_{-2}g_{-2}-1/2g_{-2}000-2h_{2}000-g_{1}g_{3}000g_{8}000g_{12}00g_{16}002g_{19}0g_{21}g_{22}0
(-1, 0, 0, 0, 0)-e_{1}+e_{2}g_{-1}0-2g_{-25}-g_{-24}0-g_{-22}00-g_{-20}00-g_{-17}000-g_{-14}000-g_{-10}0000-g_{-6}0g_{-1}-1/2g_{-1}000-2h_{1}0000g_{2}000g_{7}000g_{11}00g_{15}00g_{18}0g_{21}02g_{23}g_{24}
(0, 0, 0, 0, 0)0h_{1}-g_{-25}0g_{-23}-1/2g_{-22}1/2g_{-21}-1/2g_{-20}01/2g_{-18}-1/2g_{-17}01/2g_{-15}-1/2g_{-14}001/2g_{-11}-1/2g_{-10}001/2g_{-7}-1/2g_{-6}0001/2g_{-2}-g_{-1}00000g_{1}-1/2g_{2}0001/2g_{6}-1/2g_{7}001/2g_{10}-1/2g_{11}001/2g_{14}-1/2g_{15}01/2g_{17}-1/2g_{18}01/2g_{20}-1/2g_{21}1/2g_{22}-g_{23}0g_{25}
(0, 0, 0, 0, 0)0h_{2}0-1/2g_{-24}-g_{-23}1/2g_{-22}00g_{-19}-1/2g_{-18}01/2g_{-16}-1/2g_{-15}001/2g_{-12}-1/2g_{-11}001/2g_{-8}-1/2g_{-7}-1/2g_{-6}001/2g_{-3}-g_{-2}1/2g_{-1}00000-1/2g_{1}g_{2}-1/2g_{3}001/2g_{6}1/2g_{7}-1/2g_{8}001/2g_{11}-1/2g_{12}001/2g_{15}-1/2g_{16}01/2g_{18}-g_{19}00-1/2g_{22}g_{23}1/2g_{24}0
(0, 0, 0, 0, 0)0h_{3}000-1/2g_{-22}-1/2g_{-21}1/2g_{-20}-g_{-19}1/2g_{-18}0000g_{-13}-1/2g_{-12}0-1/2g_{-10}1/2g_{-9}-1/2g_{-8}-1/2g_{-7}1/2g_{-6}01/2g_{-4}-g_{-3}1/2g_{-2}0000000-1/2g_{2}g_{3}-1/2g_{4}0-1/2g_{6}1/2g_{7}1/2g_{8}-1/2g_{9}1/2g_{10}01/2g_{12}-g_{13}0000-1/2g_{18}g_{19}-1/2g_{20}1/2g_{21}1/2g_{22}000
(0, 0, 0, 0, 0)0h_{4}00000-1/2g_{-20}0-1/2g_{-18}1/2g_{-17}-1/2g_{-16}1/2g_{-15}-1/2g_{-14}-g_{-13}1/2g_{-12}-1/2g_{-11}1/2g_{-10}0-1/2g_{-8}1/2g_{-7}0g_{-5}-g_{-4}1/2g_{-3}000000000-1/2g_{3}g_{4}-g_{5}0-1/2g_{7}1/2g_{8}0-1/2g_{10}1/2g_{11}-1/2g_{12}g_{13}1/2g_{14}-1/2g_{15}1/2g_{16}-1/2g_{17}1/2g_{18}01/2g_{20}00000
(0, 0, 0, 0, 0)0h_{5}00000000-g_{-17}0-g_{-15}g_{-14}0-g_{-12}g_{-11}0-g_{-9}g_{-8}00-2g_{-5}g_{-4}00000000000-g_{4}2g_{5}00-g_{8}g_{9}0-g_{11}g_{12}0-g_{14}g_{15}0g_{17}00000000
(1, 0, 0, 0, 0)e_{1}-e_{2}g_{1}-g_{-24}-2g_{-23}0-g_{-21}0-g_{-18}00-g_{-15}00-g_{-11}000-g_{-7}000-g_{-2}00002h_{1}-g_{1}1/2g_{1}0000g_{6}0000g_{10}000g_{14}000g_{17}00g_{20}00g_{22}0g_{24}2g_{25}0
(0, 1, 0, 0, 0)e_{2}-e_{3}g_{2}0-g_{-22}-g_{-21}0-2g_{-19}00-g_{-16}00-g_{-12}000-g_{-8}000-g_{-3}g_{-1}0002h_{2}01/2g_{2}-g_{2}1/2g_{2}00-g_{6}0g_{7}0000g_{11}000g_{15}000g_{18}00g_{21}02g_{23}g_{24}000
(0, 0, 1, 0, 0)e_{3}-e_{4}g_{3}000-g_{-20}-g_{-18}0-g_{-16}00-2g_{-13}000-g_{-9}0g_{-6}0-g_{-4}g_{-2}0002h_{3}0001/2g_{3}-g_{3}1/2g_{3}00-g_{7}0g_{8}0-g_{10}00g_{12}000g_{16}002g_{19}0g_{21}0g_{22}00000
(0, 0, 0, 1, 0)e_{4}-e_{5}g_{4}00000-g_{-17}0-g_{-15}0-g_{-12}0g_{-10}-g_{-9}0g_{-7}0-2g_{-5}g_{-3}0002h_{4}000001/2g_{4}-g_{4}g_{4}00-g_{8}0g_{9}0-g_{11}02g_{13}-g_{14}0g_{16}00g_{18}0g_{20}00000000
(0, 0, 0, 0, 1)2e_{5}g_{5}00000000g_{-14}0g_{-11}00g_{-8}00g_{-4}000h_{5}0000000g_{5}-2g_{5}000-g_{9}000-g_{12}00-g_{15}00-g_{17}00000000000
(1, 1, 0, 0, 0)e_{1}-e_{3}g_{6}-g_{-22}-g_{-21}0-2g_{-19}0-g_{-16}00-g_{-12}00-g_{-8}000-g_{-3}0002h_{2}+2h_{1}000g_{1}-g_{2}-1/2g_{6}-1/2g_{6}1/2g_{6}0000g_{10}0000g_{14}000g_{17}000g_{20}00g_{22}0g_{24}2g_{25}000
(0, 1, 1, 0, 0)e_{2}-e_{4}g_{7}0-g_{-20}-g_{-18}0-g_{-16}00-2g_{-13}00-g_{-9}000-g_{-4}g_{-1}002h_{3}+2h_{2}000g_{2}-g_{3}01/2g_{7}-1/2g_{7}-1/2g_{7}1/2g_{7}0-g_{10}00g_{11}0000g_{15}000g_{18}00g_{21}02g_{23}0g_{24}00000
(0, 0, 1, 1, 0)e_{3}-e_{5}g_{8}000-g_{-17}-g_{-15}0-g_{-12}00-g_{-9}0g_{-6}0-2g_{-5}g_{-2}002h_{4}+2h_{3}000g_{3}-g_{4}0001/2g_{8}-1/2g_{8}-1/2g_{8}g_{8}0-g_{11}00g_{12}-g_{14}00g_{16}002g_{19}00g_{21}0g_{22}00000000
(0, 0, 0, 1, 1)e_{4}+e_{5}g_{9}00000g_{-14}0g_{-11}g_{-10}g_{-8}g_{-7}0g_{-4}g_{-3}002h_{5}+2h_{4}000g_{4}-2g_{5}000001/2g_{9}0-g_{9}00-g_{12}-2g_{13}00-g_{15}-g_{16}0-g_{17}-g_{18}00-g_{20}00000000000
(1, 1, 1, 0, 0)e_{1}-e_{4}g_{10}-g_{-20}-g_{-18}0-g_{-16}0-2g_{-13}00-g_{-9}00-g_{-4}0002h_{3}+2h_{2}+2h_{1}00g_{1}-g_{3}00g_{6}0-g_{7}-1/2g_{10}0-1/2g_{10}1/2g_{10}0000g_{14}0000g_{17}000g_{20}00g_{22}0g_{24}02g_{25}00000
(0, 1, 1, 1, 0)e_{2}-e_{5}g_{11}0-g_{-17}-g_{-15}0-g_{-12}00-g_{-9}00-2g_{-5}g_{-1}002h_{4}+2h_{3}+2h_{2}00g_{2}-g_{4}00g_{7}0-g_{8}01/2g_{11}-1/2g_{11}0-1/2g_{11}g_{11}-g_{14}000g_{15}000g_{18}00g_{21}002g_{23}0g_{24}00000000
(0, 0, 1, 1, 1)e_{3}+e_{5}g_{12}000g_{-14}g_{-11}0g_{-8}0g_{-6}g_{-4}g_{-2}002h_{5}+2h_{4}+2h_{3}00g_{3}-2g_{5}00g_{8}0-g_{9}0001/2g_{12}-1/2g_{12}1/2g_{12}-g_{12}0-g_{15}0-g_{16}0-g_{17}0-2g_{19}00-g_{21}00-g_{22}00000000000
(0, 0, 0, 2, 1)2e_{4}g_{13}00000g_{-10}0g_{-7}0g_{-3}00h_{5}+2h_{4}000g_{4}0000-g_{9}00000g_{13}-g_{13}000-g_{16}000-g_{18}00-g_{20}000000000000000
(1, 1, 1, 1, 0)e_{1}-e_{5}g_{14}-g_{-17}-g_{-15}0-g_{-12}0-g_{-9}00-2g_{-5}002h_{4}+2h_{3}+2h_{2}+2h_{1}00g_{1}-g_{4}0g_{6}0-g_{8}0g_{10}00-g_{11}-1/2g_{14}00-1/2g_{14}g_{14}0000g_{17}000g_{20}00g_{22}00g_{24}02g_{25}00000000
(0, 1, 1, 1, 1)e_{2}+e_{5}g_{15}0g_{-14}g_{-11}0g_{-8}00g_{-4}g_{-1}02h_{5}+2h_{4}+2h_{3}+2h_{2}00g_{2}-2g_{5}0g_{7}0-g_{9}0g_{11}00-g_{12}01/2g_{15}-1/2g_{15}01/2g_{15}-g_{15}-g_{17}00-g_{18}000-g_{21}00-2g_{23}00-g_{24}00000000000
(0, 0, 1, 2, 1)e_{3}+e_{4}g_{16}000g_{-10}g_{-7}g_{-6}g_{-3}g_{-2}02h_{5}+4h_{4}+2h_{3}00g_{3}g_{4}00g_{8}-g_{9}000-g_{12}-2g_{13}0001/2g_{16}0-1/2g_{16}00-g_{18}-2g_{19}00-g_{20}-g_{21}00-g_{22}000000000000000
(1, 1, 1, 1, 1)e_{1}+e_{5}g_{17}g_{-14}g_{-11}0g_{-8}0g_{-4}002h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}-2g_{5}0g_{6}0-g_{9}g_{10}00-g_{12}g_{14}000-g_{15}-1/2g_{17}001/2g_{17}-g_{17}000-g_{20}000-g_{22}00-g_{24}00-2g_{25}00000000000
(0, 1, 1, 2, 1)e_{2}+e_{4}g_{18}0g_{-10}g_{-7}0g_{-3}g_{-1}02h_{5}+4h_{4}+2h_{3}+2h_{2}0g_{2}g_{4}0g_{7}0-g_{9}0g_{11}0-2g_{13}00-g_{15}0-g_{16}01/2g_{18}-1/2g_{18}1/2g_{18}-1/2g_{18}0-g_{20}0-g_{21}000-2g_{23}00-g_{24}000000000000000
(0, 0, 2, 2, 1)2e_{3}g_{19}000g_{-6}g_{-2}0h_{5}+2h_{4}+2h_{3}00g_{3}000g_{8}000-g_{12}0000-g_{16}000g_{19}-g_{19}000-g_{21}000-g_{22}0000000000000000000
(1, 1, 1, 2, 1)e_{1}+e_{4}g_{20}g_{-10}g_{-7}0g_{-3}02h_{5}+4h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}g_{4}g_{6}0-g_{9}g_{10}00-2g_{13}g_{14}00-g_{16}0-g_{17}00-g_{18}-1/2g_{20}01/2g_{20}-1/2g_{20}000-g_{22}000-g_{24}00-2g_{25}000000000000000
(0, 1, 2, 2, 1)e_{2}+e_{3}g_{21}0g_{-6}g_{-2}g_{-1}2h_{5}+4h_{4}+4h_{3}+2h_{2}0g_{2}g_{3}0g_{7}g_{8}00g_{11}-g_{12}00-g_{15}-g_{16}000-g_{18}-2g_{19}01/2g_{21}0-1/2g_{21}00-g_{22}-2g_{23}000-g_{24}0000000000000000000
(1, 1, 2, 2, 1)e_{1}+e_{3}g_{22}g_{-6}g_{-2}02h_{5}+4h_{4}+4h_{3}+2h_{2}+2h_{1}g_{1}g_{3}g_{6}0g_{8}g_{10}0-g_{12}0g_{14}0-g_{16}0-g_{17}0-2g_{19}00-g_{20}0-g_{21}-1/2g_{22}1/2g_{22}-1/2g_{22}000-g_{24}000-2g_{25}0000000000000000000
(0, 2, 2, 2, 1)2e_{2}g_{23}0g_{-1}h_{5}+2h_{4}+2h_{3}+2h_{2}0g_{2}00g_{7}00g_{11}000-g_{15}000-g_{18}0000-g_{21}0g_{23}-g_{23}000-g_{24}000000000000000000000000
(1, 2, 2, 2, 1)e_{1}+e_{2}g_{24}g_{-1}2h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}g_{1}g_{2}g_{6}g_{7}0g_{10}g_{11}0g_{14}-g_{15}00-g_{17}-g_{18}00-g_{20}-g_{21}000-g_{22}-2g_{23}0-1/2g_{24}000-2g_{25}000000000000000000000000
(2, 2, 2, 2, 1)2e_{1}g_{25}h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}g_{1}0g_{6}0g_{10}00g_{14}00-g_{17}000-g_{20}000-g_{22}0000-g_{24}-g_{25}00000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0\\ 0 & -1/2 & 1 & -1/2 & 0\\ 0 & 0 & -1/2 & 1 & -1\\ 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0\\ 0 & -2 & 4 & -2 & 0\\ 0 & 0 & -2 & 4 & -2\\ 0 & 0 & 0 & -2 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1/8
Half sum of positive roots: (5, 9, 12, 14, 15/2)= \(\displaystyle 5\varepsilon_{1}+4\varepsilon_{2}+3\varepsilon_{3}+2\varepsilon_{4}+\varepsilon_{5}\)
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, 1/2) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 1) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
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) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1) = \(\displaystyle 2\varepsilon_{5}\)