\(F^{1}_4\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.
Page generated by the calculator project.
Lie algebra type: F^{1}_4.
Weyl group size: 1152.
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 48 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-2, -3, -4, -2)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}\)
(-1, -3, -4, -2)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{2}\)
(-1, -2, -4, -2)-e_{3}+e_{4}\(s_{3}s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}\)
(-1, -2, -3, -2)e_{4}\(s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}\)
(-1, -2, -2, -2)e_{3}+e_{4}\(s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}\)
(-1, -2, -3, -1)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}\(s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}\)
(-1, -1, -2, -2)e_{2}+e_{4}\(s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{4}\)
(-1, -2, -2, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}\(s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}\)
(0, -1, -2, -2)e_{1}+e_{4}\(s_{4}s_{3}s_{2}s_{3}s_{4}\)
(-1, -1, -2, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}\(s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}\)
(-1, -2, -2, 0)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{1}s_{3}s_{2}\)
(0, -1, -2, -1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}\(s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{3}\)
(-1, -1, -1, -1)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(-1, -1, -2, 0)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{1}s_{3}\)
(0, -1, -1, -1)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, -1, -2, 0)-e_{2}-e_{3}\(s_{3}s_{2}s_{3}\)
(-1, -1, -1, 0)-e_{1}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, -1, -1)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0)-e_{2}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}\(s_{4}\)
(0, 0, -1, 0)-e_{3}\(s_{3}\)
(0, -1, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0)e_{3}\(s_{3}\)
(0, 0, 0, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}\(s_{4}\)
(1, 1, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0)e_{2}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}\(s_{3}s_{4}s_{3}\)
(1, 1, 1, 0)e_{1}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 2, 0)e_{2}+e_{3}\(s_{3}s_{2}s_{3}\)
(0, 1, 1, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(1, 1, 2, 0)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{1}s_{3}\)
(1, 1, 1, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 2, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}\(s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{3}\)
(1, 2, 2, 0)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{1}s_{3}s_{2}\)
(1, 1, 2, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}\(s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}\)
(0, 1, 2, 2)-e_{1}-e_{4}\(s_{4}s_{3}s_{2}s_{3}s_{4}\)
(1, 2, 2, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}\(s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}\)
(1, 1, 2, 2)-e_{2}-e_{4}\(s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{4}\)
(1, 2, 3, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}\(s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}\)
(1, 2, 2, 2)-e_{3}-e_{4}\(s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}\)
(1, 2, 3, 2)-e_{4}\(s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}s_{4}\)
(1, 2, 4, 2)e_{3}-e_{4}\(s_{3}s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}\)
(1, 3, 4, 2)e_{2}-e_{4}\(s_{2}s_{3}s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{2}\)
(2, 3, 4, 2)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-2, -3, -4, -2), (-1, -3, -4, -2), (-1, -2, -4, -2), (-1, -2, -3, -2), (-1, -2, -2, -2), (-1, -2, -3, -1), (-1, -1, -2, -2), (-1, -2, -2, -1), (0, -1, -2, -2), (-1, -1, -2, -1), (-1, -2, -2, 0), (0, -1, -2, -1), (-1, -1, -1, -1), (-1, -1, -2, 0), (0, -1, -1, -1), (0, -1, -2, 0), (-1, -1, -1, 0), (0, 0, -1, -1), (0, -1, -1, 0), (-1, -1, 0, 0), (0, 0, 0, -1), (0, 0, -1, 0), (0, -1, 0, 0), (-1, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1), (1, 1, 1, 0), (0, 1, 2, 0), (0, 1, 1, 1), (1, 1, 2, 0), (1, 1, 1, 1), (0, 1, 2, 1), (1, 2, 2, 0), (1, 1, 2, 1), (0, 1, 2, 2), (1, 2, 2, 1), (1, 1, 2, 2), (1, 2, 3, 1), (1, 2, 2, 2), (1, 2, 3, 2), (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) The resulting Lie bracket pairing table follows.
Type F^{1}_4.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_{-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}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}
(-2, -3, -4, -2)-e_{1}+e_{4}g_{-24}000000000000000000000000g_{-24}000g_{-23}000g_{-22}00g_{-21}00-g_{-20}g_{-19}0g_{-18}-g_{-17}0g_{-15}-g_{-14}-g_{-12}g_{-11}-g_{-8}-g_{-5}-g_{-1}-2h_{4}-4h_{3}-3h_{2}-2h_{1}
(-1, -3, -4, -2)-e_{2}+e_{4}g_{-23}00000000000000000000000g_{-24}-g_{-23}g_{-23}000g_{-22}000g_{-21}00-g_{-20}g_{-19}00-g_{-17}g_{-16}0-g_{-14}g_{-13}0-g_{-10}g_{-9}-g_{-6}-g_{-2}-2h_{4}-4h_{3}-3h_{2}-h_{1}-g_{1}
(-1, -2, -4, -2)-e_{3}+e_{4}g_{-22}0000000000000000000g_{-24}00g_{-23}00-g_{-22}g_{-22}000g_{-21}000g_{-19}0-g_{-18}0g_{-16}0-g_{-15}0g_{-13}-g_{-11}0g_{-9}-g_{-7}0-g_{-3}-2h_{4}-4h_{3}-2h_{2}-h_{1}-g_{2}-g_{5}
(-1, -2, -3, -2)e_{4}g_{-21}00000000000000002g_{-24}02g_{-23}002g_{-22}000001/2g_{-21}002g_{-20}g_{-19}0-2g_{-18}g_{-17}2g_{-16}0-g_{-15}0g_{-13}-g_{-12}0g_{-10}-g_{-8}-g_{-7}g_{-6}-g_{-4}-g_{-3}-4h_{4}-6h_{3}-4h_{2}-2h_{1}-g_{3}-g_{6}-g_{8}
(-1, -2, -2, -2)e_{3}+e_{4}g_{-20}0000000000000-g_{-24}0-g_{-23}00000g_{-21}000g_{-20}-g_{-20}g_{-20}0g_{-18}0g_{-17}-g_{-16}0000-g_{-12}0g_{-10}000g_{-5}-g_{-4}-g_{-2}0-2h_{4}-2h_{3}-2h_{2}-h_{1}-g_{3}0g_{9}g_{11}
(-1, -2, -3, -1)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}g_{-19}0000000000002g_{-24}02g_{-23}002g_{-22}00g_{-21}000001/2g_{-19}-1/2g_{-19}00g_{-17}00-g_{-15}-2g_{-14}g_{-13}g_{-12}2g_{-11}-g_{-10}-2g_{-9}-g_{-8}g_{-7}g_{-6}0-g_{-3}0-2h_{4}-6h_{3}-4h_{2}-2h_{1}0-g_{4}-g_{7}-g_{10}-g_{12}
(-1, -1, -2, -2)e_{2}+e_{4}g_{-18}0000000000g_{-24}0000-g_{-22}00-g_{-21}000g_{-20}0g_{-18}-g_{-18}0g_{-18}g_{-16}00g_{-15}00-g_{-12}0000g_{-7}00-g_{-4}-g_{-1}0-2h_{4}-2h_{3}-h_{2}-h_{1}0-g_{2}g_{6}g_{9}0-g_{14}
(-1, -2, -2, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}g_{-17}000000000-2g_{-24}0-2g_{-23}00000g_{-21}002g_{-20}g_{-19}000g_{-17}-1/2g_{-17}00g_{-15}0-2g_{-14}-g_{-13}g_{-12}0-g_{-10}0g_{-8}0-g_{-6}2g_{-5}g_{-4}-2g_{-2}0-2h_{4}-4h_{3}-4h_{2}-2h_{1}0-g_{3}-g_{4}-g_{7}0g_{13}g_{15}
(0, -1, -2, -2)e_{1}+e_{4}g_{-16}0000000000g_{-23}00g_{-22}00g_{-21}00-g_{-20}000g_{-18}-g_{-16}00g_{-16}000g_{-13}00-g_{-10}00g_{-7}00-g_{-4}00-2h_{4}-2h_{3}-h_{2}0-g_{1}0g_{5}-g_{8}-g_{11}-g_{14}0
(-1, -1, -2, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}g_{-15}00000002g_{-24}000-2g_{-22}00-g_{-21}000-g_{-19}02g_{-18}0g_{-17}0g_{-15}-g_{-15}1/2g_{-15}0g_{-13}0g_{-12}-2g_{-11}00g_{-8}-g_{-7}00g_{-4}-g_{-3}-2g_{-1}0-2h_{4}-4h_{3}-2h_{2}-2h_{1}0-2g_{2}-g_{4}g_{6}0g_{10}g_{13}0-g_{17}
(-1, -2, -2, 0)-e_{1}-e_{2}g_{-14}000000-g_{-24}0-g_{-23}00000000-g_{-19}00-g_{-17}0000g_{-14}0-g_{-14}0g_{-11}00-g_{-9}g_{-8}0-g_{-6}g_{-5}0-g_{-2}00-2h_{3}-2h_{2}-h_{1}00g_{4}0g_{7}000g_{16}g_{18}
(0, -1, -2, -1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}g_{-13}00000002g_{-23}02g_{-22}00g_{-21}000g_{-19}00-g_{-17}2g_{-16}00g_{-15}-g_{-13}01/2g_{-13}000g_{-10}-2g_{-9}0-g_{-7}g_{-6}0g_{-4}-g_{-3}00-2h_{4}-4h_{3}-2h_{2}0-2g_{1}-g_{4}2g_{5}0-g_{8}0-g_{12}-g_{15}-g_{17}0
(-1, -1, -1, -1)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}g_{-12}00000-2g_{-24}00000-g_{-21}00-2g_{-20}g_{-19}0-2g_{-18}g_{-17}00g_{-15}00g_{-12}0-1/2g_{-12}1/2g_{-12}g_{-10}00-g_{-8}g_{-7}0-2g_{-5}g_{-4}0-2g_{-1}0-2h_{4}-2h_{3}-2h_{2}-2h_{1}00-g_{3}0-g_{6}g_{7}-2g_{9}g_{10}g_{13}00g_{19}
(-1, -1, -2, 0)-e_{1}-e_{3}g_{-11}0000g_{-24}000-g_{-22}00000g_{-19}00000-g_{-15}0g_{-14}0g_{-11}-g_{-11}g_{-11}-g_{-11}g_{-9}0g_{-8}0000-g_{-3}-g_{-1}0-2h_{3}-h_{2}-h_{1}00-g_{2}g_{4}000-g_{10}00g_{16}0-g_{20}
(0, -1, -1, -1)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}g_{-10}00000-2g_{-23}000g_{-21}002g_{-20}-g_{-19}00-g_{-17}-2g_{-16}000g_{-13}0g_{-12}-g_{-10}g_{-10}-1/2g_{-10}1/2g_{-10}0g_{-7}0-g_{-6}0g_{-4}-2g_{-2}00-2h_{4}-2h_{3}-2h_{2}0-2g_{1}-g_{3}00g_{7}g_{8}02g_{11}-g_{12}-g_{15}0g_{19}0
(0, -1, -2, 0)-e_{2}-e_{3}g_{-9}0000g_{-23}0g_{-22}00000-g_{-19}000000-g_{-14}-g_{-13}00g_{-11}-g_{-9}0g_{-9}-g_{-9}00g_{-6}00-g_{-3}00-2h_{3}-h_{2}0-g_{1}0g_{4}g_{5}0000g_{12}00-g_{18}-g_{20}0
(-1, -1, -1, 0)-e_{1}g_{-8}000-2g_{-24}0000-g_{-21}00-g_{-19}00g_{-17}00g_{-15}2g_{-14}0-g_{-12}2g_{-11}00g_{-8}00-1/2g_{-8}g_{-6}0-2g_{-5}0g_{-3}-2g_{-1}0-2h_{3}-2h_{2}-2h_{1}00-g_{3}g_{4}0-g_{6}-g_{7}0-g_{10}0g_{13}02g_{16}00g_{21}
(0, 0, -1, -1)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}g_{-7}00000-2g_{-22}0-g_{-21}00g_{-19}02g_{-18}02g_{-16}0-g_{-15}0-g_{-13}g_{-12}00g_{-10}00-g_{-7}1/2g_{-7}1/2g_{-7}00g_{-4}-g_{-3}00-2h_{4}-2h_{3}00-2g_{2}0-2g_{5}g_{6}0g_{8}-g_{10}0-g_{12}-2g_{14}0g_{17}g_{19}00
(0, -1, -1, 0)-e_{2}g_{-6}000-2g_{-23}00g_{-21}00g_{-19}00-g_{-17}000-2g_{-14}g_{-13}00-g_{-10}2g_{-9}0g_{-8}-g_{-6}g_{-6}0-1/2g_{-6}0g_{-3}-2g_{-2}00-2h_{3}-2h_{2}0-2g_{1}-g_{3}g_{4}00-g_{7}g_{8}00g_{12}0-g_{15}0-2g_{18}0g_{21}0
(-1, -1, 0, 0)-e_{1}+e_{3}g_{-5}00-g_{-24}00000g_{-20}00g_{-17}000g_{-14}0-g_{-12}000-g_{-8}00g_{-5}g_{-5}-g_{-5}0g_{-2}-g_{-1}00-h_{2}-h_{1}00g_{3}000g_{7}0-g_{9}00-g_{13}00-g_{16}000g_{22}
(0, 0, 0, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}g_{-4}00000-g_{-21}0-2g_{-20}0-2g_{-18}g_{-17}-2g_{-16}0g_{-15}0g_{-13}g_{-12}0g_{-10}00g_{-7}0000-1/2g_{-4}g_{-4}000-2h_{4}00-g_{3}00-g_{6}0-g_{8}-2g_{9}0-2g_{11}g_{13}-2g_{14}g_{15}0g_{17}g_{19}000
(0, 0, -1, 0)-e_{3}g_{-3}000-2g_{-22}-g_{-21}00-g_{-19}0000-g_{-15}0-g_{-13}0-2g_{-11}0-2g_{-9}g_{-8}-g_{-7}0g_{-6}00-g_{-3}g_{-3}-1/2g_{-3}00-2h_{3}00-2g_{2}g_{4}-2g_{5}g_{6}0g_{8}0g_{10}0g_{12}000g_{17}02g_{20}g_{21}00
(0, -1, 0, 0)-e_{2}+e_{3}g_{-2}00-g_{-23}000-g_{-20}00-g_{-17}000-g_{-14}000-g_{-10}000-g_{-6}0g_{-5}-g_{-2}2g_{-2}-g_{-2}00-h_{2}00-g_{1}g_{3}000g_{7}000g_{11}00g_{15}00g_{18}00g_{22}0
(-1, 0, 0, 0)-e_{1}+e_{2}g_{-1}0-g_{-24}000000-g_{-18}00-g_{-15}00-g_{-12}-g_{-11}00-g_{-8}000-g_{-5}02g_{-1}-g_{-1}00-h_{1}000g_{2}00g_{6}00g_{9}g_{10}00g_{13}00g_{16}00000g_{23}
(0, 0, 0, 0)0h_{1}-g_{-24}g_{-23}0000-g_{-18}0g_{-16}-g_{-15}0g_{-13}-g_{-12}-g_{-11}g_{-10}g_{-9}-g_{-8}0g_{-6}-g_{-5}00g_{-2}-2g_{-1}00002g_{1}-g_{2}00g_{5}-g_{6}0g_{8}-g_{9}-g_{10}g_{11}g_{12}-g_{13}0g_{15}-g_{16}0g_{18}0000-g_{23}g_{24}
(0, 0, 0, 0)0h_{2}0-g_{-23}g_{-22}0-g_{-20}0g_{-18}-g_{-17}0g_{-15}-g_{-14}00g_{-11}-g_{-10}00g_{-7}-g_{-6}-g_{-5}0g_{-3}-2g_{-2}g_{-1}0000-g_{1}2g_{2}-g_{3}0g_{5}g_{6}-g_{7}00g_{10}-g_{11}00g_{14}-g_{15}0g_{17}-g_{18}0g_{20}0-g_{22}g_{23}0
(0, 0, 0, 0)0h_{3}00-g_{-22}0g_{-20}-1/2g_{-19}01/2g_{-17}0-1/2g_{-15}0-1/2g_{-13}1/2g_{-12}-g_{-11}1/2g_{-10}-g_{-9}0-1/2g_{-7}0g_{-5}1/2g_{-4}-g_{-3}g_{-2}000000-g_{2}g_{3}-1/2g_{4}-g_{5}01/2g_{7}0g_{9}-1/2g_{10}g_{11}-1/2g_{12}1/2g_{13}01/2g_{15}0-1/2g_{17}01/2g_{19}-g_{20}0g_{22}00
(0, 0, 0, 0)0h_{4}000-1/2g_{-21}-g_{-20}1/2g_{-19}-g_{-18}0-g_{-16}0g_{-14}0-1/2g_{-12}g_{-11}-1/2g_{-10}g_{-9}1/2g_{-8}-1/2g_{-7}1/2g_{-6}0-g_{-4}1/2g_{-3}00000000-1/2g_{3}g_{4}0-1/2g_{6}1/2g_{7}-1/2g_{8}-g_{9}1/2g_{10}-g_{11}1/2g_{12}0-g_{14}0g_{16}0g_{18}-1/2g_{19}g_{20}1/2g_{21}000
(1, 0, 0, 0)e_{1}-e_{2}g_{1}-g_{-23}00000-g_{-16}00-g_{-13}00-g_{-10}-g_{-9}00-g_{-6}00-g_{-2}000h_{1}-2g_{1}g_{1}000g_{5}000g_{8}00g_{11}g_{12}00g_{15}00g_{18}000000g_{24}0
(0, 1, 0, 0)e_{2}-e_{3}g_{2}0-g_{-22}00-g_{-18}00-g_{-15}00-g_{-11}000-g_{-7}000-g_{-3}g_{-1}00h_{2}0g_{2}-2g_{2}g_{2}0-g_{5}0g_{6}000g_{10}000g_{14}000g_{17}00g_{20}000g_{23}00
(0, 0, 1, 0)e_{3}g_{3}00-g_{-21}-2g_{-20}0-g_{-17}000-g_{-12}0-g_{-10}0-g_{-8}0-g_{-6}2g_{-5}-g_{-4}2g_{-2}002h_{3}000g_{3}-g_{3}1/2g_{3}0-g_{6}0g_{7}-g_{8}2g_{9}02g_{11}0g_{13}0g_{15}0000g_{19}00g_{21}2g_{22}000
(0, 0, 0, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}g_{4}000-g_{-19}-g_{-17}0-g_{-15}2g_{-14}-g_{-13}2g_{-11}02g_{-9}g_{-8}0g_{-6}00g_{-3}002h_{4}000001/2g_{4}-g_{4}00-g_{7}00-g_{10}0-g_{12}-g_{13}0-g_{15}02g_{16}-g_{17}2g_{18}02g_{20}0g_{21}00000
(1, 1, 0, 0)e_{1}-e_{3}g_{5}-g_{-22}000g_{-16}00g_{-13}00g_{-9}0-g_{-7}000-g_{-3}00h_{2}+h_{1}00g_{1}-g_{2}-g_{5}-g_{5}g_{5}000g_{8}000g_{12}0-g_{14}000-g_{17}00-g_{20}00000g_{24}00
(0, 1, 1, 0)e_{2}g_{6}0-g_{-21}02g_{-18}0g_{-15}0-g_{-12}00-g_{-8}g_{-7}00-g_{-4}g_{-3}2g_{-1}02h_{3}+2h_{2}002g_{2}-g_{3}0g_{6}-g_{6}01/2g_{6}-g_{8}0-2g_{9}g_{10}00-g_{13}2g_{14}000g_{17}00-g_{19}00-g_{21}002g_{23}000
(0, 0, 1, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}g_{7}00-g_{-19}-g_{-17}02g_{-14}g_{-12}0g_{-10}-g_{-8}0-g_{-6}2g_{-5}02g_{-2}002h_{4}+2h_{3}00g_{3}-g_{4}000g_{7}-1/2g_{7}-1/2g_{7}0-g_{10}00-g_{12}g_{13}0g_{15}0-2g_{16}0-2g_{18}0-g_{19}00g_{21}02g_{22}00000
(1, 1, 1, 0)e_{1}g_{8}-g_{-21}00-2g_{-16}0-g_{-13}0g_{-10}0g_{-7}g_{-6}0-g_{-4}g_{-3}002h_{3}+2h_{2}+2h_{1}02g_{1}-g_{3}02g_{5}0-g_{6}-g_{8}001/2g_{8}00-2g_{11}g_{12}0-2g_{14}-g_{15}00-g_{17}00g_{19}00g_{21}00002g_{24}000
(0, 1, 2, 0)e_{2}+e_{3}g_{9}0g_{-20}g_{-18}00-g_{-12}0000-g_{-5}-g_{-4}0g_{-1}02h_{3}+h_{2}00g_{3}00-g_{6}00g_{9}0-g_{9}g_{9}-g_{11}00g_{13}g_{14}000000g_{19}00000-g_{22}0-g_{23}0000
(0, 1, 1, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}g_{10}0-g_{-19}0g_{-15}g_{-12}-2g_{-11}0-g_{-8}-g_{-7}00g_{-3}2g_{-1}02h_{4}+2h_{3}+2h_{2}002g_{2}-g_{4}0g_{6}0-g_{7}0g_{10}-g_{10}1/2g_{10}-1/2g_{10}-g_{12}0-g_{13}0002g_{16}g_{17}00g_{19}-2g_{20}00-g_{21}0002g_{23}00000
(1, 1, 2, 0)e_{1}+e_{3}g_{11}g_{-20}0-g_{-16}00g_{-10}000-g_{-4}g_{-2}002h_{3}+h_{2}+h_{1}0g_{1}g_{3}0000-g_{8}0-g_{9}-g_{11}g_{11}-g_{11}g_{11}0-g_{14}0g_{15}00000-g_{19}00000g_{22}000-g_{24}0000
(1, 1, 1, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}g_{12}-g_{-19}00-g_{-13}-g_{-10}2g_{-9}-g_{-7}g_{-6}0g_{-3}002h_{4}+2h_{3}+2h_{2}+2h_{1}02g_{1}0-g_{4}2g_{5}0-g_{7}g_{8}00-g_{10}-g_{12}01/2g_{12}-1/2g_{12}00-g_{15}00-g_{17}2g_{18}0-g_{19}2g_{20}00g_{21}000002g_{24}00000
(0, 1, 2, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}g_{13}0g_{-17}g_{-15}g_{-12}0g_{-8}0-2g_{-5}g_{-4}2g_{-1}02h_{4}+4h_{3}+2h_{2}00g_{3}-g_{4}0-g_{6}g_{7}02g_{9}-g_{10}00g_{13}0-1/2g_{13}0-g_{15}00-2g_{16}g_{17}00-g_{19}000-g_{21}00-2g_{22}0-2g_{23}0000000
(1, 2, 2, 0)e_{1}+e_{2}g_{14}-g_{-18}-g_{-16}000-g_{-7}0-g_{-4}002h_{3}+2h_{2}+h_{1}00g_{2}0-g_{5}g_{6}0-g_{8}g_{9}00-g_{11}00-g_{14}0g_{14}000g_{17}00g_{19}00000000g_{23}0g_{24}000000
(1, 1, 2, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}g_{15}g_{-17}0-g_{-13}-g_{-10}0-g_{-6}g_{-4}2g_{-2}02h_{4}+4h_{3}+2h_{2}+2h_{1}02g_{1}g_{3}-g_{4}00g_{7}-g_{8}002g_{11}-g_{12}0-g_{13}-g_{15}g_{15}-1/2g_{15}00-g_{17}0-2g_{18}0g_{19}000g_{21}002g_{22}000-2g_{24}0000000
(0, 1, 2, 2)-e_{1}-e_{4}g_{16}0g_{-14}g_{-11}g_{-8}-g_{-5}0g_{-1}02h_{4}+2h_{3}+h_{2}00g_{4}00-g_{7}00g_{10}00-g_{13}000g_{16}00-g_{16}-g_{18}000g_{20}00-g_{21}00-g_{22}00-g_{23}0000000000
(1, 2, 2, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}g_{17}-g_{-15}-g_{-13}0g_{-7}g_{-4}g_{-3}02h_{4}+4h_{3}+4h_{2}+2h_{1}02g_{2}-g_{4}-2g_{5}g_{6}0-g_{8}0g_{10}0-g_{12}g_{13}2g_{14}0-g_{15}00-g_{17}1/2g_{17}000-g_{19}-2g_{20}00-g_{21}000002g_{23}02g_{24}000000000
(1, 1, 2, 2)-e_{2}-e_{4}g_{18}g_{-14}0-g_{-9}-g_{-6}g_{-2}02h_{4}+2h_{3}+h_{2}+h_{1}0g_{1}g_{4}00-g_{7}0000g_{12}00-g_{15}00-g_{16}-g_{18}g_{18}0-g_{18}0-g_{20}000g_{21}00g_{22}0000-g_{24}0000000000
(1, 2, 3, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}g_{19}g_{-12}g_{-10}g_{-7}g_{-4}02h_{4}+6h_{3}+4h_{2}+2h_{1}0g_{3}0-g_{6}-g_{7}g_{8}2g_{9}g_{10}-2g_{11}-g_{12}-g_{13}2g_{14}g_{15}00-g_{17}0000-1/2g_{19}1/2g_{19}000-g_{21}00-2g_{22}00-2g_{23}0-2g_{24}000000000000
(1, 2, 2, 2)-e_{3}-e_{4}g_{20}-g_{-11}-g_{-9}0g_{-3}2h_{4}+2h_{3}+2h_{2}+h_{1}0g_{2}g_{4}-g_{5}000-g_{10}0g_{12}0000g_{16}-g_{17}0-g_{18}00-g_{20}g_{20}-g_{20}00-g_{21}00000g_{23}0g_{24}0000000000000
(1, 2, 3, 2)-e_{4}g_{21}g_{-8}g_{-6}g_{-3}4h_{4}+6h_{3}+4h_{2}+2h_{1}g_{3}g_{4}-g_{6}g_{7}g_{8}-g_{10}0g_{12}-g_{13}0g_{15}0-2g_{16}-g_{17}2g_{18}0-g_{19}-2g_{20}00000-1/2g_{21}00-2g_{22}00-2g_{23}0-2g_{24}0000000000000000
(1, 2, 4, 2)e_{3}-e_{4}g_{22}g_{-5}g_{-2}2h_{4}+4h_{3}+2h_{2}+h_{1}g_{3}0g_{7}-g_{9}0g_{11}-g_{13}0g_{15}0-g_{16}0g_{18}0-g_{19}000-g_{21}000g_{22}-g_{22}00-g_{23}00-g_{24}0000000000000000000
(1, 3, 4, 2)e_{2}-e_{4}g_{23}g_{-1}2h_{4}+4h_{3}+3h_{2}+h_{1}g_{2}g_{6}-g_{9}g_{10}0-g_{13}g_{14}0-g_{16}g_{17}00-g_{19}g_{20}00-g_{21}000-g_{22}0g_{23}-g_{23}00-g_{24}00000000000000000000000
(2, 3, 4, 2)e_{1}-e_{4}g_{24}2h_{4}+4h_{3}+3h_{2}+2h_{1}g_{1}g_{5}g_{8}-g_{11}g_{12}g_{14}-g_{15}0g_{17}-g_{18}0-g_{19}g_{20}00-g_{21}00-g_{22}000-g_{23}-g_{24}000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -1 & 0\\ 0 & -1 & 1 & -1/2\\ 0 & 0 & -1/2 & 1\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -2 & 0\\ 0 & -2 & 4 & -2\\ 0 & 0 & -2 & 4\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1/4
Half sum of positive roots: (8, 15, 21, 11)= \(\displaystyle 5/2\varepsilon_{1}+3/2\varepsilon_{2}+1/2\varepsilon_{3}-11/2\varepsilon_{4}\)
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):
(2, 3, 4, 2) = \(\displaystyle \varepsilon_{1}-\varepsilon_{4}\)
(3, 6, 8, 4) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}-2\varepsilon_{4}\)
(2, 4, 6, 3) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-3/2\varepsilon_{4}\)
(1, 2, 3, 2) = \(\displaystyle -\varepsilon_{4}\)
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) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0) = \(\displaystyle \varepsilon_{3}\)
(0, 0, 0, 1) = \(\displaystyle -1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}\)