\(E^{1}_7\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: E^{1}_7.
Weyl group size: 2903040.
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 126 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-2, -2, -3, -4, -3, -2, -1)-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)-1/2e_{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)1/2e_{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)1/2e_{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, -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}\(s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}\)
(-1, -2, -2, -3, -2, -2, -1)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, -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}\(s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}\)
(-1, -2, -2, -3, -2, -1, -1)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, -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}\(s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{6}\)
(-1, -1, -2, -3, -2, -1, -1)-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)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, -1, -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}\(s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}\)
(-1, -1, -2, -2, -2, -1, -1)-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)-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, -2, -1)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)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)-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)-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, -2, -1, -1)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)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)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)-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, -2, -1, -1, -1)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)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)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)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)-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, -1, -1, -1, -1, -1, -1)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)-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)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)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)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, -1, -1, -1, -1, -1)-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)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)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)-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)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)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, -1, -1, -1, -1)-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)-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)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)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)-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)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, -1, -1, -1)-e_{3}+e_{6}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, 0)-e_{2}+e_{5}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0)-e_{1}+e_{4}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, 0, -1, -1, 0, 0)e_{1}+e_{4}\(s_{2}s_{4}s_{5}s_{4}s_{2}\)
(0, -1, -1, -1, 0, 0, 0)e_{2}+e_{3}\(s_{2}s_{3}s_{4}s_{2}s_{3}\)
(-1, 0, -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}\(s_{1}s_{3}s_{4}s_{3}s_{1}\)
(0, 0, 0, 0, 0, -1, -1)-e_{4}+e_{6}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, 0)-e_{3}+e_{5}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0)-e_{2}+e_{4}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0)-e_{1}+e_{3}\(s_{3}s_{4}s_{3}\)
(0, -1, 0, -1, 0, 0, 0)e_{1}+e_{3}\(s_{2}s_{4}s_{2}\)
(-1, 0, -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}\(s_{1}s_{3}s_{1}\)
(0, 0, 0, 0, 0, 0, -1)-e_{5}+e_{6}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0)-e_{4}+e_{5}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0)-e_{3}+e_{4}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0)-e_{2}+e_{3}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0)e_{1}+e_{2}\(s_{2}\)
(-1, 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)-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)-e_{1}-e_{2}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0)e_{1}-e_{2}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0)e_{2}-e_{3}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0)e_{3}-e_{4}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0)e_{4}-e_{5}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1)e_{5}-e_{6}\(s_{7}\)
(1, 0, 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}\(s_{1}s_{3}s_{1}\)
(0, 1, 0, 1, 0, 0, 0)-e_{1}-e_{3}\(s_{2}s_{4}s_{2}\)
(0, 0, 1, 1, 0, 0, 0)e_{1}-e_{3}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0)e_{2}-e_{4}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0)e_{3}-e_{5}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1)e_{4}-e_{6}\(s_{6}s_{7}s_{6}\)
(1, 0, 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}\(s_{1}s_{3}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 0, 0, 0)-e_{2}-e_{3}\(s_{2}s_{3}s_{4}s_{2}s_{3}\)
(0, 1, 0, 1, 1, 0, 0)-e_{1}-e_{4}\(s_{2}s_{4}s_{5}s_{4}s_{2}\)
(0, 0, 1, 1, 1, 0, 0)e_{1}-e_{4}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0)e_{2}-e_{5}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1)e_{3}-e_{6}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(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}\(s_{1}s_{2}s_{3}s_{4}s_{2}s_{3}s_{1}\)
(1, 0, 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}\(s_{1}s_{3}s_{4}s_{5}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 1, 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)-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)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)e_{2}-e_{6}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(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}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 1, 1, 2, 1, 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)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)-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)-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)e_{1}-e_{6}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 1, 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}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(1, 1, 1, 1, 1, 1, 0)-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)-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)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)-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, 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}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 1, 1, 0)-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)-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)-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)-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, 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}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 1, 0)-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)-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)-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, 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}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(1, 1, 2, 2, 1, 1, 1)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)-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)-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, 2, 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}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(1, 1, 2, 2, 2, 1, 1)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)-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, 2, 2, 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}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(1, 1, 2, 3, 2, 1, 1)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)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, 2, 2, 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}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(1, 1, 2, 3, 2, 2, 1)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, 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}\(s_{2}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}\)
(1, 1, 2, 3, 3, 2, 1)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, 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}\(s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}\)
(1, 2, 2, 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}\(s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}\)
(1, 2, 3, 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}\(s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(2, 2, 3, 4, 3, 2, 1)e_{7}-e_{8}\(s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}\)
Comma delimited list of roots: (-2, -2, -3, -4, -3, -2, -1), (-1, -2, -3, -4, -3, -2, -1), (-1, -2, -2, -4, -3, -2, -1), (-1, -2, -2, -3, -3, -2, -1), (-1, -1, -2, -3, -3, -2, -1), (-1, -2, -2, -3, -2, -2, -1), (-1, -1, -2, -3, -2, -2, -1), (-1, -2, -2, -3, -2, -1, -1), (-1, -1, -2, -2, -2, -2, -1), (-1, -1, -2, -3, -2, -1, -1), (-1, -2, -2, -3, -2, -1, 0), (-1, -1, -1, -2, -2, -2, -1), (-1, -1, -2, -2, -2, -1, -1), (-1, -1, -2, -3, -2, -1, 0), (0, -1, -1, -2, -2, -2, -1), (-1, -1, -1, -2, -2, -1, -1), (-1, -1, -2, -2, -1, -1, -1), (-1, -1, -2, -2, -2, -1, 0), (0, -1, -1, -2, -2, -1, -1), (-1, -1, -1, -2, -1, -1, -1), (-1, -1, -1, -2, -2, -1, 0), (-1, -1, -2, -2, -1, -1, 0), (0, -1, -1, -2, -1, -1, -1), (-1, -1, -1, -1, -1, -1, -1), (0, -1, -1, -2, -2, -1, 0), (-1, -1, -1, -2, -1, -1, 0), (-1, -1, -2, -2, -1, 0, 0), (0, -1, -1, -1, -1, -1, -1), (-1, 0, -1, -1, -1, -1, -1), (0, -1, -1, -2, -1, -1, 0), (-1, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -2, -1, 0, 0), (0, 0, -1, -1, -1, -1, -1), (0, -1, 0, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, 0), (-1, 0, -1, -1, -1, -1, 0), (0, -1, -1, -2, -1, 0, 0), (-1, -1, -1, -1, -1, 0, 0), (0, 0, 0, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, 0), (0, -1, 0, -1, -1, -1, 0), (0, -1, -1, -1, -1, 0, 0), (-1, 0, -1, -1, -1, 0, 0), (-1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, -1, -1, -1), (0, 0, 0, -1, -1, -1, 0), (0, 0, -1, -1, -1, 0, 0), (0, -1, 0, -1, -1, 0, 0), (0, -1, -1, -1, 0, 0, 0), (-1, 0, -1, -1, 0, 0, 0), (0, 0, 0, 0, 0, -1, -1), (0, 0, 0, 0, -1, -1, 0), (0, 0, 0, -1, -1, 0, 0), (0, 0, -1, -1, 0, 0, 0), (0, -1, 0, -1, 0, 0, 0), (-1, 0, -1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, -1), (0, 0, 0, 0, 0, -1, 0), (0, 0, 0, 0, -1, 0, 0), (0, 0, 0, -1, 0, 0, 0), (0, 0, -1, 0, 0, 0, 0), (0, -1, 0, 0, 0, 0, 0), (-1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1), (1, 0, 1, 0, 0, 0, 0), (0, 1, 0, 1, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 1, 1), (1, 0, 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, 0, 1, 1, 1, 0), (0, 0, 0, 0, 1, 1, 1), (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, 1, 1, 1, 1), (1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 2, 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, 1, 1, 1, 1, 1), (1, 1, 1, 2, 1, 0, 0), (1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 2, 1, 1, 0), (1, 0, 1, 1, 1, 1, 1), (0, 1, 1, 1, 1, 1, 1), (1, 1, 2, 2, 1, 0, 0), (1, 1, 1, 2, 1, 1, 0), (0, 1, 1, 2, 2, 1, 0), (1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 2, 1, 1, 1), (1, 1, 2, 2, 1, 1, 0), (1, 1, 1, 2, 2, 1, 0), (1, 1, 1, 2, 1, 1, 1), (0, 1, 1, 2, 2, 1, 1), (1, 1, 2, 2, 2, 1, 0), (1, 1, 2, 2, 1, 1, 1), (1, 1, 1, 2, 2, 1, 1), (0, 1, 1, 2, 2, 2, 1), (1, 1, 2, 3, 2, 1, 0), (1, 1, 2, 2, 2, 1, 1), (1, 1, 1, 2, 2, 2, 1), (1, 2, 2, 3, 2, 1, 0), (1, 1, 2, 3, 2, 1, 1), (1, 1, 2, 2, 2, 2, 1), (1, 2, 2, 3, 2, 1, 1), (1, 1, 2, 3, 2, 2, 1), (1, 2, 2, 3, 2, 2, 1), (1, 1, 2, 3, 3, 2, 1), (1, 2, 2, 3, 3, 2, 1), (1, 2, 2, 4, 3, 2, 1), (1, 2, 3, 4, 3, 2, 1), (2, 2, 3, 4, 3, 2, 1) The resulting Lie bracket pairing table follows.
Type E^{1}_7.The letter \(\displaystyle h\) stands for elements of the Cartan subalgebra,
the letter \(\displaystyle g\) stands for the Chevalley (root space) generators of non-zero weight.
The generator \(\displaystyle h_i\) is the element of the Cartan subalgebra dual to the
i^th simple root, that is, \(\displaystyle [h_i, g] =\langle \alpha_i , \gamma\rangle g\),
where g is a Chevalley generator, \(\displaystyle \gamma\) is its weight, and
\(\displaystyle \alpha_i\) is the i^th simple root.
The Lie bracket table is too large to be rendered in LaTeX, displaying in html format instead.
roots simple coords epsilon coordinates[,]g_{-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}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}
(-2, -2, -3, -4, -3, -2, -1)-e_{7}+e_{8}g_{-63}000000000000000000000000000000000000000000000000000000000000000g_{-63}000000g_{-62}000000g_{-61}00000g_{-60}00000-g_{-59}g_{-58}0000-g_{-57}0g_{-56}000g_{-55}-g_{-54}0-g_{-53}0-g_{-52}g_{-51}0g_{-50}0-g_{-48}-g_{-47}-g_{-46}0g_{-44}g_{-43}g_{-42}0-g_{-40}-g_{-38}g_{-37}g_{-35}g_{-33}-g_{-32}-g_{-28}g_{-26}-g_{-21}g_{-20}-g_{-14}-g_{-8}-g_{-1}-h_{7}-2h_{6}-3h_{5}-4h_{4}-3h_{3}-2h_{2}-2h_{1}
(-1, -2, -3, -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}g_{-62}00000000000000000000000000000000000000000000000000000000000000g_{-63}-g_{-62}0g_{-62}000000g_{-61}000000g_{-60}0000-g_{-59}0g_{-58}0000-g_{-57}0g_{-56}00g_{-55}0-g_{-54}0-g_{-53}00g_{-51}0g_{-50}-g_{-49}0-g_{-47}0-g_{-46}-g_{-45}00g_{-42}g_{-41}g_{-39}0g_{-37}-g_{-36}-g_{-34}0g_{-31}g_{-29}-g_{-27}-g_{-24}g_{-22}-g_{-17}g_{-15}-g_{-10}-g_{-3}-h_{7}-2h_{6}-3h_{5}-4h_{4}-3h_{3}-2h_{2}-h_{1}-g_{1}
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(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}g_{46}-g_{-44}-g_{-41}0g_{-30}g_{-25}000-g_{-13}0g_{-9}0-g_{-7}g_{-4}000h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}00g_{3}g_{5}00-g_{8}0-g_{12}000-g_{17}000g_{21}g_{22}0g_{24}0-g_{26}0-g_{28}-g_{29}000g_{33}0000g_{37}000g_{39}00-g_{42}0-g_{43}0000-g_{46}g_{46}-g_{46}0g_{46}000-g_{50}00g_{51}0-g_{53}000g_{55}00000000000-g_{59}0000-g_{60}0000000000g_{62}00g_{63}0000000000000000000
(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}g_{47}-g_{-43}-g_{-39}000-g_{-23}-g_{-18}g_{-16}-g_{-12}g_{-11}00g_{-5}000h_{7}+h_{6}+h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}00g_{3}0-g_{7}-g_{8}g_{10}00-g_{13}-g_{14}-g_{15}000g_{20}0000000000-g_{31}0000g_{35}g_{36}g_{37}00-g_{40}0g_{41}g_{42}000-g_{44}0000-g_{47}0g_{47}0-g_{47}0000-g_{51}00000-g_{54}g_{55}000-g_{56}0g_{57}0000g_{58}000000000000000g_{62}000g_{63}00000000000000000000
(1, 1, 1, 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}g_{48}-g_{-42}0g_{-34}g_{-29}g_{-24}00-g_{-15}0-g_{-10}0g_{-6}g_{-3}00h_{7}+h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}00g_{1}g_{5}-g_{7}00-g_{11}0000g_{16}00-g_{19}0g_{21}000g_{25}-g_{26}000-g_{30}0g_{32}00-g_{35}0000g_{40}000g_{43}0-g_{44}000-g_{45}-g_{48}0g_{48}0-g_{48}g_{48}-g_{48}00-g_{51}00-g_{52}000g_{54}0000g_{56}00000000-g_{59}0000-g_{60}0000-g_{61}0000000g_{63}000000000000000000000
(0, 1, 1, 2, 2, 2, 1)-e_{5}-e_{6}g_{49}0-g_{-37}-g_{-32}-g_{-26}-g_{-21}g_{-20}g_{-14}0-g_{-8}00g_{-1}00h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}000g_{6}000-g_{12}0-g_{13}00g_{18}0g_{19}00-g_{23}-g_{24}-g_{25}000g_{29}g_{30}g_{31}000-g_{34}-g_{36}0000g_{39}g_{41}00000-g_{45}00000g_{49}0000-g_{49}0-g_{52}000000g_{55}00000-g_{57}00000-g_{58}g_{59}0000g_{60}00000g_{61}0000g_{62}00000000000000000000000000
(1, 1, 2, 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}g_{50}g_{-40}g_{-36}g_{-30}0-g_{-19}0-g_{-13}00-g_{-7}g_{-2}00h_{6}+2h_{5}+3h_{4}+2h_{3}+h_{2}+h_{1}000g_{4}00-g_{10}g_{11}00g_{14}-g_{17}-g_{18}00g_{21}0g_{24}000g_{27}-g_{28}00-g_{32}00-g_{34}00g_{37}g_{38}00-g_{39}00-g_{42}g_{43}00000-g_{46}0000g_{50}0-g_{50}00g_{50}0-g_{53}0000g_{54}00000g_{57}00000g_{59}0000000000-g_{61}00000-g_{62}000-g_{63}00000000000000000000000
(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}g_{51}g_{-38}g_{-34}0-g_{-23}-g_{-18}00g_{-9}g_{-6}g_{-4}00h_{7}+h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}00g_{3}g_{5}-g_{7}-g_{8}0000-g_{17}00-g_{19}g_{21}g_{22}000-g_{26}0000g_{31}000-g_{35}-g_{36}0g_{37}0g_{40}00000000g_{45}g_{46}0-g_{47}0-g_{48}0000-g_{51}g_{51}-g_{51}g_{51}-g_{51}000-g_{54}0-g_{55}00-g_{56}00000000g_{59}0000g_{60}0000000000-g_{62}000-g_{63}0000000000000000000000000
(1, 1, 1, 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}g_{52}-g_{-37}0g_{-27}g_{-22}g_{-17}-g_{-15}-g_{-10}0g_{-3}00h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}00g_{1}g_{6}000-g_{12}-g_{13}00g_{18}0g_{19}00-g_{23}0-g_{25}00-g_{28}0g_{30}00g_{33}0g_{35}000-g_{38}-g_{40}0000g_{43}g_{44}00000-g_{48}0000-g_{49}-g_{52}0g_{52}00-g_{52}000-g_{55}000000g_{57}0000g_{58}0-g_{59}0000-g_{60}0000-g_{61}000000000g_{63}00000000000000000000000000
(1, 2, 2, 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}g_{53}-g_{-35}-g_{-31}-g_{-25}-g_{-19}0-g_{-13}0-g_{-7}00h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}00g_{2}000g_{9}00-g_{15}g_{16}00g_{20}-g_{22}-g_{23}00g_{26}g_{27}g_{29}00-g_{32}0-g_{33}-g_{34}00g_{37}g_{38}0-g_{39}000-g_{42}g_{43}00000-g_{46}000000-g_{50}00-g_{53}0000g_{53}000000g_{56}00000g_{58}00000g_{60}00000g_{61}00000g_{62}000g_{63}0000000000000000000000000000
(1, 1, 2, 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}g_{54}-g_{-33}-g_{-29}-g_{-23}0g_{-12}0g_{-6}g_{-2}0h_{7}+h_{6}+2h_{5}+3h_{4}+2h_{3}+h_{2}+h_{1}00g_{4}-g_{7}0-g_{10}g_{11}0g_{14}-g_{17}00g_{21}000-g_{25}0g_{27}00g_{31}-g_{32}000-g_{35}0g_{37}000-g_{41}000g_{44}00-g_{45}00-g_{47}g_{48}00g_{50}00-g_{51}0000g_{54}0-g_{54}0g_{54}-g_{54}0-g_{56}000-g_{57}00000-g_{59}0000000000g_{61}00000g_{62}000g_{63}000000000000000000000000000000
(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}g_{55}g_{-32}g_{-27}0-g_{-16}-g_{-11}g_{-9}g_{-4}0h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}00g_{3}g_{6}0-g_{8}0-g_{12}-g_{13}000g_{19}0g_{24}000-g_{28}-g_{29}0-g_{31}0g_{33}0g_{35}g_{36}000-g_{40}0000-g_{42}00000g_{46}g_{47}000g_{49}0-g_{51}00-g_{52}0000-g_{55}g_{55}0-g_{55}0000-g_{57}0000-g_{58}0g_{59}0000g_{60}0000000000-g_{62}0000-g_{63}0000000000000000000000000000000
(1, 2, 2, 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}g_{56}g_{-28}g_{-24}g_{-18}g_{-12}0g_{-6}0h_{7}+h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}0g_{2}-g_{7}0g_{9}00-g_{15}g_{16}0g_{20}-g_{22}00g_{26}g_{27}00-g_{30}-g_{32}000g_{36}0g_{37}00-g_{40}-g_{41}000g_{44}0-g_{45}000-g_{47}g_{48}00000-g_{51}0g_{53}0000-g_{54}00-g_{56}000g_{56}-g_{56}00000-g_{58}00000-g_{60}00000-g_{61}00000-g_{62}000-g_{63}00000000000000000000000000000000000
(1, 1, 2, 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}g_{57}-g_{-26}-g_{-22}-g_{-16}0g_{-5}g_{-2}h_{7}+2h_{6}+2h_{5}+3h_{4}+2h_{3}+h_{2}+h_{1}0g_{4}g_{6}0-g_{10}0-g_{13}g_{14}0-g_{18}00g_{24}0g_{25}-g_{28}00-g_{31}00-g_{34}g_{35}00g_{38}00g_{41}00-g_{42}-g_{44}00000g_{47}000-g_{49}g_{50}00g_{52}000-g_{54}0-g_{55}0000g_{57}0-g_{57}g_{57}-g_{57}00-g_{58}00-g_{59}0000000000g_{61}00000g_{62}000g_{63}0000000000000000000000000000000000000
(1, 2, 2, 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}g_{58}g_{-21}g_{-17}g_{-11}g_{-5}0h_{7}+2h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}g_{2}g_{6}g_{9}0-g_{13}-g_{15}00g_{20}0-g_{23}00g_{29}0g_{30}-g_{33}-g_{34}0-g_{36}0g_{38}0g_{40}g_{41}00-g_{42}-g_{44}00000g_{47}00-g_{49}0000g_{52}0g_{53}000-g_{55}00-g_{56}000-g_{57}00-g_{58}00g_{58}-g_{58}00000-g_{60}00000-g_{61}00000-g_{62}000-g_{63}000000000000000000000000000000000000000000
(1, 1, 2, 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}g_{59}-g_{-20}-g_{-15}-g_{-9}g_{-2}h_{7}+2h_{6}+3h_{5}+3h_{4}+2h_{3}+h_{2}+h_{1}0g_{5}0-g_{11}g_{12}0g_{17}-g_{18}-g_{19}-g_{21}g_{24}0g_{25}-g_{28}0-g_{31}000g_{35}000-g_{39}000g_{43}00g_{45}00-g_{46}-g_{48}00g_{49}0g_{50}g_{51}-g_{52}0000-g_{54}g_{55}00000-g_{57}00000g_{59}00-g_{59}000-g_{60}000000g_{61}00000g_{62}0000g_{63}0000000000000000000000000000000000000000000
(1, 2, 2, 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}g_{60}g_{-14}g_{-10}g_{-4}h_{7}+2h_{6}+3h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}g_{2}g_{5}0g_{12}-g_{16}0-g_{19}g_{22}-g_{23}0-g_{26}g_{29}0g_{30}-g_{33}0-g_{36}00-g_{39}g_{40}00g_{43}00g_{45}00-g_{46}-g_{48}00g_{49}00g_{51}-g_{52}00g_{53}00g_{55}000-g_{56}000000-g_{58}00-g_{59}00-g_{60}0g_{60}-g_{60}00000-g_{61}00000-g_{62}000-g_{63}0000000000000000000000000000000000000000000000000
(1, 2, 2, 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}g_{61}g_{-8}g_{-3}h_{7}+2h_{6}+3h_{5}+4h_{4}+2h_{3}+2h_{2}+h_{1}g_{4}-g_{9}g_{11}-g_{16}g_{18}0-g_{23}-g_{25}g_{27}0g_{30}-g_{32}g_{34}00-g_{38}-g_{39}-g_{41}0g_{43}0g_{44}g_{45}000-g_{48}0g_{49}0-g_{50}00-g_{52}0g_{53}0g_{54}0000-g_{56}0g_{57}0000-g_{58}0g_{59}0000-g_{60}00000g_{61}-g_{61}00000-g_{62}0000-g_{63}0000000000000000000000000000000000000000000000000000000
(1, 2, 3, 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}g_{62}g_{-1}h_{7}+2h_{6}+3h_{5}+4h_{4}+3h_{3}+2h_{2}+h_{1}g_{3}g_{10}-g_{15}g_{17}-g_{22}g_{24}g_{27}-g_{29}-g_{31}0g_{34}g_{36}-g_{37}0-g_{39}-g_{41}-g_{42}00g_{45}g_{46}0g_{47}0g_{49}-g_{50}0-g_{51}00g_{53}0g_{54}0-g_{55}00-g_{56}0g_{57}0000-g_{58}0g_{59}0000-g_{60}000000-g_{61}00g_{62}0-g_{62}0000-g_{63}00000000000000000000000000000000000000000000000000000000000000
(2, 2, 3, 4, 3, 2, 1)e_{7}-e_{8}g_{63}h_{7}+2h_{6}+3h_{5}+4h_{4}+3h_{3}+2h_{2}+2h_{1}g_{1}g_{8}g_{14}-g_{20}g_{21}-g_{26}g_{28}g_{32}-g_{33}-g_{35}-g_{37}g_{38}g_{40}0-g_{42}-g_{43}-g_{44}0g_{46}g_{47}g_{48}0-g_{50}0-g_{51}g_{52}0g_{53}0g_{54}-g_{55}000-g_{56}0g_{57}0000-g_{58}g_{59}00000-g_{60}00000-g_{61}000000-g_{62}-g_{63}000000000000000000000000000000000000000000000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Let the (i, j)^{th} entry of the symmetric Cartan matrix be a_{ij}.
Then we define the co-symmetric Cartan matrix as the matrix whose (i, j)^{th} entry equals 4*a_{ij}/(a_{ii}*a_{jj}). In other words, the co-symmetric Cartan matrix is the symmetric Cartan matrix of the dual root system. The co-symmetric Cartan matrix equals:
\(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 2
Half sum of positive roots: (17, 49/2, 33, 48, 75/2, 26, 27/2)= \(\displaystyle -\varepsilon_{2}-2\varepsilon_{3}-3\varepsilon_{4}-4\varepsilon_{5}-5\varepsilon_{6}+17/2\varepsilon_{7}-17/2\varepsilon_{8}\)
The fundamental weights (the j^th fundamental weight has scalar product 1
with the j^th simple root times 2 divided by the root length squared,
and 0 with the remaining simple roots):
(2, 2, 3, 4, 3, 2, 1) = \(\displaystyle \varepsilon_{7}-\varepsilon_{8}\)
(2, 7/2, 4, 6, 9/2, 3, 3/2) = \(\displaystyle -1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}+\varepsilon_{7}-\varepsilon_{8}\)
(3, 4, 6, 8, 6, 4, 2) = \(\displaystyle 1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}+3/2\varepsilon_{7}-3/2\varepsilon_{8}\)
(4, 6, 8, 12, 9, 6, 3) = \(\displaystyle -\varepsilon_{3}-\varepsilon_{4}-\varepsilon_{5}-\varepsilon_{6}+2\varepsilon_{7}-2\varepsilon_{8}\)
(3, 9/2, 6, 9, 15/2, 5, 5/2) = \(\displaystyle -\varepsilon_{4}-\varepsilon_{5}-\varepsilon_{6}+3/2\varepsilon_{7}-3/2\varepsilon_{8}\)
(2, 3, 4, 6, 5, 4, 2) = \(\displaystyle -\varepsilon_{5}-\varepsilon_{6}+\varepsilon_{7}-\varepsilon_{8}\)
(1, 3/2, 2, 3, 5/2, 2, 3/2) = \(\displaystyle -\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}\)
Below is the simple basis realized in epsilon coordinates. Please note that the epsilon coordinate realizations do not have long roots of length of 2 in types G and C. This means that gramm matrix (w.r.t. the standard scalar product) of the epsilon coordinate realizations in types G and C does not equal the corresponding symmetric Cartan matrix.
(1, 0, 0, 0, 0, 0, 0) = \(\displaystyle -1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}\)
(0, 1, 0, 0, 0, 0, 0) = \(\displaystyle -\varepsilon_{1}-\varepsilon_{2}\)
(0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)