\(E^{1}_8\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: E^{1}_8.
Weyl group size: 696729600.
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 240 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-2, -3, -4, -6, -5, -4, -3, -2)e_{7}+e_{8}\(s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}\)
(-2, -3, -4, -6, -5, -4, -3, -1)e_{6}+e_{8}\(s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}\)
(-2, -3, -4, -6, -5, -4, -2, -1)e_{5}+e_{8}\(s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}\)
(-2, -3, -4, -6, -5, -3, -2, -1)e_{4}+e_{8}\(s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(-2, -3, -4, -6, -4, -3, -2, -1)e_{3}+e_{8}\(s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(-2, -3, -4, -5, -4, -3, -2, -1)e_{2}+e_{8}\(s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(-2, -2, -4, -5, -4, -3, -2, -1)-e_{1}+e_{8}\(s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(-2, -3, -3, -5, -4, -3, -2, -1)e_{1}+e_{8}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(-1, -3, -3, -5, -4, -3, -2, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(-2, -2, -3, -5, -4, -3, -2, -1)-e_{2}+e_{8}\(s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(-1, -2, -3, -5, -4, -3, -2, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(-2, -2, -3, -4, -4, -3, -2, -1)-e_{3}+e_{8}\(s_{1}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}\)
(-1, -2, -3, -4, -4, -3, -2, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(-2, -2, -3, -4, -3, -3, -2, -1)-e_{4}+e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}\)
(-1, -2, -2, -4, -4, -3, -2, -1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(-1, -2, -3, -4, -3, -3, -2, -1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(-2, -2, -3, -4, -3, -2, -2, -1)-e_{5}+e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}\)
(-1, -2, -2, -4, -3, -3, -2, -1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(-1, -2, -3, -4, -3, -2, -2, -1)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(-2, -2, -3, -4, -3, -2, -1, -1)-e_{6}+e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(-1, -2, -2, -3, -3, -3, -2, -1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{7}s_{6}\)
(-1, -2, -2, -4, -3, -2, -2, -1)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{7}\)
(-1, -2, -3, -4, -3, -2, -1, -1)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(-2, -2, -3, -4, -3, -2, -1, 0)-e_{7}+e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(-1, -1, -2, -3, -3, -3, -2, -1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}\)
(-1, -2, -2, -3, -3, -2, -2, -1)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{7}\)
(-1, -2, -2, -4, -3, -2, -1, -1)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}\)
(-1, -2, -3, -4, -3, -2, -1, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(-1, -1, -2, -3, -3, -2, -2, -1)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}\)
(-1, -2, -2, -3, -2, -2, -2, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{7}\)
(-1, -2, -2, -3, -3, -2, -1, -1)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}\)
(-1, -2, -2, -4, -3, -2, -1, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}\)
(-1, -1, -2, -3, -2, -2, -2, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{7}\)
(-1, -1, -2, -3, -3, -2, -1, -1)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}\)
(-1, -2, -2, -3, -2, -2, -1, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}\)
(-1, -2, -2, -3, -3, -2, -1, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}\)
(-1, -1, -2, -2, -2, -2, -2, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{6}s_{7}\)
(-1, -1, -2, -3, -2, -2, -1, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}\)
(-1, -2, -2, -3, -2, -1, -1, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(-1, -1, -2, -3, -3, -2, -1, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}\)
(-1, -2, -2, -3, -2, -2, -1, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}\)
(-1, -1, -1, -2, -2, -2, -2, -1)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}s_{7}\)
(-1, -1, -2, -2, -2, -2, -1, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{6}\)
(-1, -1, -2, -3, -2, -1, -1, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(-1, -1, -2, -3, -2, -2, -1, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}\)
(-1, -2, -2, -3, -2, -1, -1, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(0, -1, -1, -2, -2, -2, -2, -1)e_{6}+e_{7}\(s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}\)
(-1, -1, -1, -2, -2, -2, -1, -1)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}\)
(-1, -1, -2, -2, -2, -1, -1, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(-1, -1, -2, -2, -2, -2, -1, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{6}\)
(-1, -1, -2, -3, -2, -1, -1, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(-1, -2, -2, -3, -2, -1, 0, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(0, -1, -1, -2, -2, -2, -1, -1)e_{5}+e_{7}\(s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}\)
(-1, -1, -1, -2, -2, -1, -1, -1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(-1, -1, -2, -2, -1, -1, -1, -1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(-1, -1, -1, -2, -2, -2, -1, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}\)
(-1, -1, -2, -2, -2, -1, -1, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(-1, -1, -2, -3, -2, -1, 0, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(0, -1, -1, -2, -2, -1, -1, -1)e_{4}+e_{7}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(-1, -1, -1, -2, -1, -1, -1, -1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(0, -1, -1, -2, -2, -2, -1, 0)e_{5}+e_{6}\(s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}\)
(-1, -1, -1, -2, -2, -1, -1, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(-1, -1, -2, -2, -1, -1, -1, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(-1, -1, -2, -2, -2, -1, 0, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(0, -1, -1, -2, -1, -1, -1, -1)e_{3}+e_{7}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(-1, -1, -1, -1, -1, -1, -1, -1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, -1, -1, -2, -2, -1, -1, 0)e_{4}+e_{6}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(-1, -1, -1, -2, -1, -1, -1, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(-1, -1, -1, -2, -2, -1, 0, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(-1, -1, -2, -2, -1, -1, 0, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(0, -1, -1, -1, -1, -1, -1, -1)e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, -1, -1, -1, -1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, -1, -1, -2, -1, -1, -1, 0)e_{3}+e_{6}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(-1, -1, -1, -1, -1, -1, -1, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, -1, -1, -2, -2, -1, 0, 0)e_{4}+e_{5}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(-1, -1, -1, -2, -1, -1, 0, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(-1, -1, -2, -2, -1, 0, 0, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(0, 0, -1, -1, -1, -1, -1, -1)-e_{1}+e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, 0, -1, -1, -1, -1, -1)e_{1}+e_{7}\(s_{2}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(0, -1, -1, -1, -1, -1, -1, 0)e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, -1, -1, -1, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, -1, -1, -2, -1, -1, 0, 0)e_{3}+e_{5}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(-1, -1, -1, -1, -1, -1, 0, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(-1, -1, -1, -2, -1, 0, 0, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(0, 0, 0, -1, -1, -1, -1, -1)-e_{2}+e_{7}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, -1, 0)-e_{1}+e_{6}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, 0, -1, -1, -1, -1, 0)e_{1}+e_{6}\(s_{2}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(0, -1, -1, -1, -1, -1, 0, 0)e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, -1, -1, 0, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, -1, -1, -2, -1, 0, 0, 0)e_{3}+e_{4}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(-1, -1, -1, -1, -1, 0, 0, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 0, 0, 0, -1, -1, -1, -1)-e_{3}+e_{7}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, -1, 0)-e_{2}+e_{6}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, 0, 0)-e_{1}+e_{5}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, 0, -1, -1, -1, 0, 0)e_{1}+e_{5}\(s_{2}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}\)
(0, -1, -1, -1, -1, 0, 0, 0)e_{2}+e_{4}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, -1, 0, 0, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{4}s_{3}s_{1}\)
(-1, -1, -1, -1, 0, 0, 0, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{2}s_{3}s_{1}\)
(0, 0, 0, 0, 0, -1, -1, -1)-e_{4}+e_{7}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, -1, 0)-e_{3}+e_{6}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, 0, 0)-e_{2}+e_{5}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0, 0)-e_{1}+e_{4}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, 0, -1, -1, 0, 0, 0)e_{1}+e_{4}\(s_{2}s_{4}s_{5}s_{4}s_{2}\)
(0, -1, -1, -1, 0, 0, 0, 0)e_{2}+e_{3}\(s_{2}s_{3}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, 0, 0, 0, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{3}s_{1}\)
(0, 0, 0, 0, 0, 0, -1, -1)-e_{5}+e_{7}\(s_{7}s_{8}s_{7}\)
(0, 0, 0, 0, 0, -1, -1, 0)-e_{4}+e_{6}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, 0, 0)-e_{3}+e_{5}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0, 0)-e_{2}+e_{4}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{3}s_{4}s_{3}\)
(0, -1, 0, -1, 0, 0, 0, 0)e_{1}+e_{3}\(s_{2}s_{4}s_{2}\)
(-1, 0, -1, 0, 0, 0, 0, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{1}\)
(0, 0, 0, 0, 0, 0, 0, -1)-e_{6}+e_{7}\(s_{8}\)
(0, 0, 0, 0, 0, 0, -1, 0)-e_{5}+e_{6}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0, 0)-e_{4}+e_{5}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0, 0)-e_{3}+e_{4}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0, 0)e_{1}+e_{2}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0, 0)-e_{1}-e_{2}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0, 0)e_{2}-e_{3}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0, 0)e_{3}-e_{4}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0, 0)e_{4}-e_{5}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 0)e_{5}-e_{6}\(s_{7}\)
(0, 0, 0, 0, 0, 0, 0, 1)e_{6}-e_{7}\(s_{8}\)
(1, 0, 1, 0, 0, 0, 0, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{1}\)
(0, 1, 0, 1, 0, 0, 0, 0)-e_{1}-e_{3}\(s_{2}s_{4}s_{2}\)
(0, 0, 1, 1, 0, 0, 0, 0)e_{1}-e_{3}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0, 0)e_{2}-e_{4}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0, 0)e_{3}-e_{5}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 0)e_{4}-e_{6}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 1)e_{5}-e_{7}\(s_{7}s_{8}s_{7}\)
(1, 0, 1, 1, 0, 0, 0, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 0, 0, 0, 0)-e_{2}-e_{3}\(s_{2}s_{3}s_{4}s_{2}s_{3}\)
(0, 1, 0, 1, 1, 0, 0, 0)-e_{1}-e_{4}\(s_{2}s_{4}s_{5}s_{4}s_{2}\)
(0, 0, 1, 1, 1, 0, 0, 0)e_{1}-e_{4}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0, 0)e_{2}-e_{5}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 0)e_{3}-e_{6}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 1)e_{4}-e_{7}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(1, 1, 1, 1, 0, 0, 0, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{2}s_{3}s_{1}\)
(1, 0, 1, 1, 1, 0, 0, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 1, 0, 0, 0)-e_{2}-e_{4}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}\)
(0, 1, 0, 1, 1, 1, 0, 0)-e_{1}-e_{5}\(s_{2}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}\)
(0, 0, 1, 1, 1, 1, 0, 0)e_{1}-e_{5}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 0)e_{2}-e_{6}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 1)e_{3}-e_{7}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 0, 0, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 1, 1, 2, 1, 0, 0, 0)-e_{3}-e_{4}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(1, 0, 1, 1, 1, 1, 0, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 1, 1, 0, 0)-e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(0, 1, 0, 1, 1, 1, 1, 0)-e_{1}-e_{6}\(s_{2}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 0)e_{1}-e_{6}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 1)e_{2}-e_{7}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 2, 1, 0, 0, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(1, 1, 1, 1, 1, 1, 0, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 1, 1, 2, 1, 1, 0, 0)-e_{3}-e_{5}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(1, 0, 1, 1, 1, 1, 1, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 0)-e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(0, 1, 0, 1, 1, 1, 1, 1)-e_{1}-e_{7}\(s_{2}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 1)e_{1}-e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 1, 0, 0, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 1, 1, 0, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(0, 1, 1, 2, 2, 1, 0, 0)-e_{4}-e_{5}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(1, 1, 1, 1, 1, 1, 1, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 1, 1, 2, 1, 1, 1, 0)-e_{3}-e_{6}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(1, 0, 1, 1, 1, 1, 1, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 1)-e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(1, 1, 2, 2, 1, 1, 0, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 1, 0, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(1, 1, 1, 2, 1, 1, 1, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(0, 1, 1, 2, 2, 1, 1, 0)-e_{4}-e_{6}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(1, 1, 1, 1, 1, 1, 1, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 1, 1, 2, 1, 1, 1, 1)-e_{3}-e_{7}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(1, 1, 2, 2, 2, 1, 0, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(1, 1, 2, 2, 1, 1, 1, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 1, 1, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(0, 1, 1, 2, 2, 2, 1, 0)-e_{5}-e_{6}\(s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}\)
(1, 1, 1, 2, 1, 1, 1, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(0, 1, 1, 2, 2, 1, 1, 1)-e_{4}-e_{7}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(1, 1, 2, 3, 2, 1, 0, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(1, 1, 2, 2, 2, 1, 1, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(1, 1, 1, 2, 2, 2, 1, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}\)
(1, 1, 2, 2, 1, 1, 1, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 1, 1, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(0, 1, 1, 2, 2, 2, 1, 1)-e_{5}-e_{7}\(s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}\)
(1, 2, 2, 3, 2, 1, 0, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(1, 1, 2, 3, 2, 1, 1, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(1, 1, 2, 2, 2, 2, 1, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{6}\)
(1, 1, 2, 2, 2, 1, 1, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(1, 1, 1, 2, 2, 2, 1, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}\)
(0, 1, 1, 2, 2, 2, 2, 1)-e_{6}-e_{7}\(s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}\)
(1, 2, 2, 3, 2, 1, 1, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(1, 1, 2, 3, 2, 2, 1, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}\)
(1, 1, 2, 3, 2, 1, 1, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(1, 1, 2, 2, 2, 2, 1, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{6}\)
(1, 1, 1, 2, 2, 2, 2, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}s_{7}\)
(1, 2, 2, 3, 2, 2, 1, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}\)
(1, 1, 2, 3, 3, 2, 1, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}\)
(1, 2, 2, 3, 2, 1, 1, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(1, 1, 2, 3, 2, 2, 1, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}\)
(1, 1, 2, 2, 2, 2, 2, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{6}s_{7}\)
(1, 2, 2, 3, 3, 2, 1, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}\)
(1, 2, 2, 3, 2, 2, 1, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}\)
(1, 1, 2, 3, 3, 2, 1, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}\)
(1, 1, 2, 3, 2, 2, 2, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{7}\)
(1, 2, 2, 4, 3, 2, 1, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}\)
(1, 2, 2, 3, 3, 2, 1, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}\)
(1, 2, 2, 3, 2, 2, 2, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{7}\)
(1, 1, 2, 3, 3, 2, 2, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}\)
(1, 2, 3, 4, 3, 2, 1, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(1, 2, 2, 4, 3, 2, 1, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}\)
(1, 2, 2, 3, 3, 2, 2, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{7}\)
(1, 1, 2, 3, 3, 3, 2, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}\)
(2, 2, 3, 4, 3, 2, 1, 0)e_{7}-e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(1, 2, 3, 4, 3, 2, 1, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(1, 2, 2, 4, 3, 2, 2, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{7}\)
(1, 2, 2, 3, 3, 3, 2, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{7}s_{6}\)
(2, 2, 3, 4, 3, 2, 1, 1)e_{6}-e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(1, 2, 3, 4, 3, 2, 2, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(1, 2, 2, 4, 3, 3, 2, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(2, 2, 3, 4, 3, 2, 2, 1)e_{5}-e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}\)
(1, 2, 3, 4, 3, 3, 2, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(1, 2, 2, 4, 4, 3, 2, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(2, 2, 3, 4, 3, 3, 2, 1)e_{4}-e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}\)
(1, 2, 3, 4, 4, 3, 2, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(2, 2, 3, 4, 4, 3, 2, 1)e_{3}-e_{8}\(s_{1}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}\)
(1, 2, 3, 5, 4, 3, 2, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(2, 2, 3, 5, 4, 3, 2, 1)e_{2}-e_{8}\(s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(1, 3, 3, 5, 4, 3, 2, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(2, 3, 3, 5, 4, 3, 2, 1)-e_{1}-e_{8}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}\)
(2, 2, 4, 5, 4, 3, 2, 1)e_{1}-e_{8}\(s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(2, 3, 4, 5, 4, 3, 2, 1)-e_{2}-e_{8}\(s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(2, 3, 4, 6, 4, 3, 2, 1)-e_{3}-e_{8}\(s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(2, 3, 4, 6, 5, 3, 2, 1)-e_{4}-e_{8}\(s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(2, 3, 4, 6, 5, 4, 2, 1)-e_{5}-e_{8}\(s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}\)
(2, 3, 4, 6, 5, 4, 3, 1)-e_{6}-e_{8}\(s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}\)
(2, 3, 4, 6, 5, 4, 3, 2)-e_{7}-e_{8}\(s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}\)
Comma delimited list of roots: (-2, -3, -4, -6, -5, -4, -3, -2), (-2, -3, -4, -6, -5, -4, -3, -1), (-2, -3, -4, -6, -5, -4, -2, -1), (-2, -3, -4, -6, -5, -3, -2, -1), (-2, -3, -4, -6, -4, -3, -2, -1), (-2, -3, -4, -5, -4, -3, -2, -1), (-2, -2, -4, -5, -4, -3, -2, -1), (-2, -3, -3, -5, -4, -3, -2, -1), (-1, -3, -3, -5, -4, -3, -2, -1), (-2, -2, -3, -5, -4, -3, -2, -1), (-1, -2, -3, -5, -4, -3, -2, -1), (-2, -2, -3, -4, -4, -3, -2, -1), (-1, -2, -3, -4, -4, -3, -2, -1), (-2, -2, -3, -4, -3, -3, -2, -1), (-1, -2, -2, -4, -4, -3, -2, -1), (-1, -2, -3, -4, -3, -3, -2, -1), (-2, -2, -3, -4, -3, -2, -2, -1), (-1, -2, -2, -4, -3, -3, -2, -1), (-1, -2, -3, -4, -3, -2, -2, -1), (-2, -2, -3, -4, -3, -2, -1, -1), (-1, -2, -2, -3, -3, -3, -2, -1), (-1, -2, -2, -4, -3, -2, -2, -1), (-1, -2, -3, -4, -3, -2, -1, -1), (-2, -2, -3, -4, -3, -2, -1, 0), (-1, -1, -2, -3, -3, -3, -2, -1), (-1, -2, -2, -3, -3, -2, -2, -1), (-1, -2, -2, -4, -3, -2, -1, -1), (-1, -2, -3, -4, -3, -2, -1, 0), (-1, -1, -2, -3, -3, -2, -2, -1), (-1, -2, -2, -3, -2, -2, -2, -1), (-1, -2, -2, -3, -3, -2, -1, -1), (-1, -2, -2, -4, -3, -2, -1, 0), (-1, -1, -2, -3, -2, -2, -2, -1), (-1, -1, -2, -3, -3, -2, -1, -1), (-1, -2, -2, -3, -2, -2, -1, -1), (-1, -2, -2, -3, -3, -2, -1, 0), (-1, -1, -2, -2, -2, -2, -2, -1), (-1, -1, -2, -3, -2, -2, -1, -1), (-1, -2, -2, -3, -2, -1, -1, -1), (-1, -1, -2, -3, -3, -2, -1, 0), (-1, -2, -2, -3, -2, -2, -1, 0), (-1, -1, -1, -2, -2, -2, -2, -1), (-1, -1, -2, -2, -2, -2, -1, -1), (-1, -1, -2, -3, -2, -1, -1, -1), (-1, -1, -2, -3, -2, -2, -1, 0), (-1, -2, -2, -3, -2, -1, -1, 0), (0, -1, -1, -2, -2, -2, -2, -1), (-1, -1, -1, -2, -2, -2, -1, -1), (-1, -1, -2, -2, -2, -1, -1, -1), (-1, -1, -2, -2, -2, -2, -1, 0), (-1, -1, -2, -3, -2, -1, -1, 0), (-1, -2, -2, -3, -2, -1, 0, 0), (0, -1, -1, -2, -2, -2, -1, -1), (-1, -1, -1, -2, -2, -1, -1, -1), (-1, -1, -2, -2, -1, -1, -1, -1), (-1, -1, -1, -2, -2, -2, -1, 0), (-1, -1, -2, -2, -2, -1, -1, 0), (-1, -1, -2, -3, -2, -1, 0, 0), (0, -1, -1, -2, -2, -1, -1, -1), (-1, -1, -1, -2, -1, -1, -1, -1), (0, -1, -1, -2, -2, -2, -1, 0), (-1, -1, -1, -2, -2, -1, -1, 0), (-1, -1, -2, -2, -1, -1, -1, 0), (-1, -1, -2, -2, -2, -1, 0, 0), (0, -1, -1, -2, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, -1, -1), (0, -1, -1, -2, -2, -1, -1, 0), (-1, -1, -1, -2, -1, -1, -1, 0), (-1, -1, -1, -2, -2, -1, 0, 0), (-1, -1, -2, -2, -1, -1, 0, 0), (0, -1, -1, -1, -1, -1, -1, -1), (-1, 0, -1, -1, -1, -1, -1, -1), (0, -1, -1, -2, -1, -1, -1, 0), (-1, -1, -1, -1, -1, -1, -1, 0), (0, -1, -1, -2, -2, -1, 0, 0), (-1, -1, -1, -2, -1, -1, 0, 0), (-1, -1, -2, -2, -1, 0, 0, 0), (0, 0, -1, -1, -1, -1, -1, -1), (0, -1, 0, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, -1, 0), (-1, 0, -1, -1, -1, -1, -1, 0), (0, -1, -1, -2, -1, -1, 0, 0), (-1, -1, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -2, -1, 0, 0, 0), (0, 0, 0, -1, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, -1, 0), (0, -1, 0, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, -1, 0, 0), (-1, 0, -1, -1, -1, -1, 0, 0), (0, -1, -1, -2, -1, 0, 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, 0, -1, -1, -1, 0, 0), (0, -1, -1, -1, -1, 0, 0, 0), (-1, 0, -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, 0, -1, -1, 0, 0, 0), (0, -1, -1, -1, 0, 0, 0, 0), (-1, 0, -1, -1, 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, 0, -1, 0, 0, 0, 0), (-1, 0, -1, 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, 0, 1, 0, 0, 0, 0, 0), (0, 1, 0, 1, 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, 0, 1, 1, 0, 0, 0, 0), (0, 1, 1, 1, 0, 0, 0, 0), (0, 1, 0, 1, 1, 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), (1, 0, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 0, 0, 0), (0, 1, 0, 1, 1, 1, 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, 2, 1, 0, 0, 0), (1, 0, 1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 1, 0, 0), (0, 1, 0, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1, 1, 1), (1, 1, 1, 2, 1, 0, 0, 0), (1, 1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 2, 1, 1, 0, 0), (1, 0, 1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1, 1, 0), (0, 1, 0, 1, 1, 1, 1, 1), (0, 0, 1, 1, 1, 1, 1, 1), (1, 1, 2, 2, 1, 0, 0, 0), (1, 1, 1, 2, 1, 1, 0, 0), (0, 1, 1, 2, 2, 1, 0, 0), (1, 1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 2, 1, 1, 1, 0), (1, 0, 1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 1, 1, 1, 1), (1, 1, 2, 2, 1, 1, 0, 0), (1, 1, 1, 2, 2, 1, 0, 0), (1, 1, 1, 2, 1, 1, 1, 0), (0, 1, 1, 2, 2, 1, 1, 0), (1, 1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 2, 1, 1, 1, 1), (1, 1, 2, 2, 2, 1, 0, 0), (1, 1, 2, 2, 1, 1, 1, 0), (1, 1, 1, 2, 2, 1, 1, 0), (0, 1, 1, 2, 2, 2, 1, 0), (1, 1, 1, 2, 1, 1, 1, 1), (0, 1, 1, 2, 2, 1, 1, 1), (1, 1, 2, 3, 2, 1, 0, 0), (1, 1, 2, 2, 2, 1, 1, 0), (1, 1, 1, 2, 2, 2, 1, 0), (1, 1, 2, 2, 1, 1, 1, 1), (1, 1, 1, 2, 2, 1, 1, 1), (0, 1, 1, 2, 2, 2, 1, 1), (1, 2, 2, 3, 2, 1, 0, 0), (1, 1, 2, 3, 2, 1, 1, 0), (1, 1, 2, 2, 2, 2, 1, 0), (1, 1, 2, 2, 2, 1, 1, 1), (1, 1, 1, 2, 2, 2, 1, 1), (0, 1, 1, 2, 2, 2, 2, 1), (1, 2, 2, 3, 2, 1, 1, 0), (1, 1, 2, 3, 2, 2, 1, 0), (1, 1, 2, 3, 2, 1, 1, 1), (1, 1, 2, 2, 2, 2, 1, 1), (1, 1, 1, 2, 2, 2, 2, 1), (1, 2, 2, 3, 2, 2, 1, 0), (1, 1, 2, 3, 3, 2, 1, 0), (1, 2, 2, 3, 2, 1, 1, 1), (1, 1, 2, 3, 2, 2, 1, 1), (1, 1, 2, 2, 2, 2, 2, 1), (1, 2, 2, 3, 3, 2, 1, 0), (1, 2, 2, 3, 2, 2, 1, 1), (1, 1, 2, 3, 3, 2, 1, 1), (1, 1, 2, 3, 2, 2, 2, 1), (1, 2, 2, 4, 3, 2, 1, 0), (1, 2, 2, 3, 3, 2, 1, 1), (1, 2, 2, 3, 2, 2, 2, 1), (1, 1, 2, 3, 3, 2, 2, 1), (1, 2, 3, 4, 3, 2, 1, 0), (1, 2, 2, 4, 3, 2, 1, 1), (1, 2, 2, 3, 3, 2, 2, 1), (1, 1, 2, 3, 3, 3, 2, 1), (2, 2, 3, 4, 3, 2, 1, 0), (1, 2, 3, 4, 3, 2, 1, 1), (1, 2, 2, 4, 3, 2, 2, 1), (1, 2, 2, 3, 3, 3, 2, 1), (2, 2, 3, 4, 3, 2, 1, 1), (1, 2, 3, 4, 3, 2, 2, 1), (1, 2, 2, 4, 3, 3, 2, 1), (2, 2, 3, 4, 3, 2, 2, 1), (1, 2, 3, 4, 3, 3, 2, 1), (1, 2, 2, 4, 4, 3, 2, 1), (2, 2, 3, 4, 3, 3, 2, 1), (1, 2, 3, 4, 4, 3, 2, 1), (2, 2, 3, 4, 4, 3, 2, 1), (1, 2, 3, 5, 4, 3, 2, 1), (2, 2, 3, 5, 4, 3, 2, 1), (1, 3, 3, 5, 4, 3, 2, 1), (2, 3, 3, 5, 4, 3, 2, 1), (2, 2, 4, 5, 4, 3, 2, 1), (2, 3, 4, 5, 4, 3, 2, 1), (2, 3, 4, 6, 4, 3, 2, 1), (2, 3, 4, 6, 5, 3, 2, 1), (2, 3, 4, 6, 5, 4, 2, 1), (2, 3, 4, 6, 5, 4, 3, 1), (2, 3, 4, 6, 5, 4, 3, 2) The resulting Lie bracket pairing table follows.
Type E^{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_{-120}g_{-119}g_{-118}g_{-117}g_{-116}g_{-115}g_{-114}g_{-113}g_{-112}g_{-111}g_{-110}g_{-109}g_{-108}g_{-107}g_{-106}g_{-105}g_{-104}g_{-103}g_{-102}g_{-101}g_{-100}g_{-99}g_{-98}g_{-97}g_{-96}g_{-95}g_{-94}g_{-93}g_{-92}g_{-91}g_{-90}g_{-89}g_{-88}g_{-87}g_{-86}g_{-85}g_{-84}g_{-83}g_{-82}g_{-81}g_{-80}g_{-79}g_{-78}g_{-77}g_{-76}g_{-75}g_{-74}g_{-73}g_{-72}g_{-71}g_{-70}g_{-69}g_{-68}g_{-67}g_{-66}g_{-65}g_{-64}g_{-63}g_{-62}g_{-61}g_{-60}g_{-59}g_{-58}g_{-57}g_{-56}g_{-55}g_{-54}g_{-53}g_{-52}g_{-51}g_{-50}g_{-49}g_{-48}g_{-47}g_{-46}g_{-45}g_{-44}g_{-43}g_{-42}g_{-41}g_{-40}g_{-39}g_{-38}g_{-37}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}g_{37}g_{38}g_{39}g_{40}g_{41}g_{42}g_{43}g_{44}g_{45}g_{46}g_{47}g_{48}g_{49}g_{50}g_{51}g_{52}g_{53}g_{54}g_{55}g_{56}g_{57}g_{58}g_{59}g_{60}g_{61}g_{62}g_{63}g_{64}g_{65}g_{66}g_{67}g_{68}g_{69}g_{70}g_{71}g_{72}g_{73}g_{74}g_{75}g_{76}g_{77}g_{78}g_{79}g_{80}g_{81}g_{82}g_{83}g_{84}g_{85}g_{86}g_{87}g_{88}g_{89}g_{90}g_{91}g_{92}g_{93}g_{94}g_{95}g_{96}g_{97}g_{98}g_{99}g_{100}g_{101}g_{102}g_{103}g_{104}g_{105}g_{106}g_{107}g_{108}g_{109}g_{110}g_{111}g_{112}g_{113}g_{114}g_{115}g_{116}g_{117}g_{118}g_{119}g_{120}
(-2, -3, -4, -6, -5, -4, -3, -2)e_{7}+e_{8}g_{-120}0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000g_{-120}0000000g_{-119}000000-g_{-118}000000g_{-117}000000-g_{-116}000000g_{-115}00000-g_{-114}-g_{-113}00000g_{-112}g_{-111}0000-g_{-110}-g_{-109}0000g_{-108}g_{-107}000-g_{-106}-g_{-105}-g_{-104}000g_{-103}g_{-102}g_{-101}00-g_{-100}-g_{-99}-g_{-98}00g_{-96}g_{-95}g_{-94}0-g_{-92}-g_{-91}-g_{-90}0g_{-88}g_{-87}g_{-86}0-g_{-84}-g_{-83}-g_{-82}0g_{-79}g_{-78}g_{-77}-g_{-74}-g_{-73}-g_{-72}g_{-68}g_{-67}g_{-66}-g_{-62}-g_{-61}g_{-56}g_{-55}-g_{-50}-g_{-49}g_{-43}g_{-42}-g_{-36}g_{-29}-g_{-22}g_{-15}-g_{-8}-2h_{8}-3h_{7}-4h_{6}-5h_{5}-6h_{4}-4h_{3}-3h_{2}-2h_{1}
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(2, 3, 4, 6, 5, 4, 2, 1)-e_{5}-e_{8}g_{118}-g_{-15}g_{-7}h_{8}+2h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}g_{6}-g_{13}g_{20}-g_{26}-g_{27}g_{32}g_{33}-g_{38}-g_{39}g_{45}g_{46}-g_{51}-g_{52}0g_{57}0g_{60}-g_{63}0-g_{65}-g_{68}g_{69}0g_{71}g_{73}00-g_{76}-g_{78}0g_{80}g_{81}g_{83}0-g_{85}0-g_{86}-g_{87}0g_{89}0g_{90}00-g_{93}0-g_{94}0-g_{96}g_{97}00g_{98}0g_{100}00-g_{101}00-g_{103}0000g_{105}g_{106}0000-g_{107}-g_{108}00000g_{109}g_{110}0000-g_{111}-g_{112}0000g_{113}g_{114}00000-g_{115}000000g_{116}000000-g_{117}0000000000-g_{118}g_{118}0000000-g_{119}0000000g_{120}000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
(2, 3, 4, 6, 5, 4, 3, 1)-e_{6}-e_{8}g_{119}g_{-8}h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}g_{7}-g_{14}g_{21}-g_{28}g_{34}g_{35}-g_{40}-g_{41}g_{47}g_{48}-g_{53}-g_{54}g_{58}g_{59}g_{60}-g_{64}-g_{65}0g_{70}g_{71}0-g_{74}-g_{75}-g_{76}0g_{79}g_{80}g_{81}0-g_{84}-g_{85}00g_{88}g_{89}00-g_{91}-g_{92}-g_{93}00g_{95}g_{96}g_{97}00-g_{99}-g_{100}0000g_{102}g_{103}000-g_{104}-g_{105}-g_{106}000g_{107}g_{108}0000-g_{109}-g_{110}00000g_{111}g_{112}0000-g_{113}-g_{114}00000g_{115}000000-g_{116}000000g_{117}000000-g_{118}000000000000-g_{119}g_{119}0000000-g_{120}0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
(2, 3, 4, 6, 5, 4, 3, 2)-e_{7}-e_{8}g_{120}2h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}g_{8}-g_{15}g_{22}-g_{29}g_{36}-g_{42}-g_{43}g_{49}g_{50}-g_{55}-g_{56}g_{61}g_{62}-g_{66}-g_{67}-g_{68}g_{72}g_{73}g_{74}-g_{77}-g_{78}-g_{79}0g_{82}g_{83}g_{84}0-g_{86}-g_{87}-g_{88}0g_{90}g_{91}g_{92}0-g_{94}-g_{95}-g_{96}00g_{98}g_{99}g_{100}00-g_{101}-g_{102}-g_{103}000g_{104}g_{105}g_{106}000-g_{107}-g_{108}0000g_{109}g_{110}0000-g_{111}-g_{112}00000g_{113}g_{114}00000-g_{115}000000g_{116}000000-g_{117}000000g_{118}000000-g_{119}00000000000000-g_{120}000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1
Half sum of positive roots: (46, 68, 91, 135, 110, 84, 57, 29)= \(\displaystyle -\varepsilon_{2}-2\varepsilon_{3}-3\varepsilon_{4}-4\varepsilon_{5}-5\varepsilon_{6}-6\varepsilon_{7}-23\varepsilon_{8}\)
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):
(4, 5, 7, 10, 8, 6, 4, 2) = \(\displaystyle -2\varepsilon_{8}\)
(5, 8, 10, 15, 12, 9, 6, 3) = \(\displaystyle -1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}-5/2\varepsilon_{8}\)
(7, 10, 14, 20, 16, 12, 8, 4) = \(\displaystyle 1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}-7/2\varepsilon_{8}\)
(10, 15, 20, 30, 24, 18, 12, 6) = \(\displaystyle -\varepsilon_{3}-\varepsilon_{4}-\varepsilon_{5}-\varepsilon_{6}-\varepsilon_{7}-5\varepsilon_{8}\)
(8, 12, 16, 24, 20, 15, 10, 5) = \(\displaystyle -\varepsilon_{4}-\varepsilon_{5}-\varepsilon_{6}-\varepsilon_{7}-4\varepsilon_{8}\)
(6, 9, 12, 18, 15, 12, 8, 4) = \(\displaystyle -\varepsilon_{5}-\varepsilon_{6}-\varepsilon_{7}-3\varepsilon_{8}\)
(4, 6, 8, 12, 10, 8, 6, 3) = \(\displaystyle -\varepsilon_{6}-\varepsilon_{7}-2\varepsilon_{8}\)
(2, 3, 4, 6, 5, 4, 3, 2) = \(\displaystyle -\varepsilon_{7}-\varepsilon_{8}\)
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, 0) = \(\displaystyle -1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}\)
(0, 1, 0, 0, 0, 0, 0, 0) = \(\displaystyle -\varepsilon_{1}-\varepsilon_{2}\)
(0, 0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)