sp(16), type \(C^{1}_8\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: C^{1}_8.
Weyl group size: 10321920.
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 128 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-2, -2, -2, -2, -2, -2, -2, -1)-2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(-1, -2, -2, -2, -2, -2, -2, -1)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, -2, -2, -2, -2, -2, -2, -1)-2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -2, -2, -2, -1)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -2, -2, -2, -2, -2, -1)-e_{2}-e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(-1, -1, -1, -2, -2, -2, -2, -1)-e_{1}-e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -2, -2, -2, -2, -2, -1)-2e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -2, -2, -2, -2, -1)-e_{2}-e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(-1, -1, -1, -1, -2, -2, -2, -1)-e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, -1, -2, -2, -2, -2, -1)-e_{3}-e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, -1, -1, -1, -2, -2, -2, -1)-e_{2}-e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}s_{5}\)
(-1, -1, -1, -1, -1, -2, -2, -1)-e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}\)
(0, 0, 0, -2, -2, -2, -2, -1)-2e_{4}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -2, -2, -2, -1)-e_{3}-e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}s_{6}s_{5}\)
(0, -1, -1, -1, -1, -2, -2, -1)-e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}\)
(-1, -1, -1, -1, -1, -1, -2, -1)-e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}\)
(0, 0, 0, -1, -2, -2, -2, -1)-e_{4}-e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, -1, -1, -1, -2, -2, -1)-e_{3}-e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}s_{6}\)
(0, -1, -1, -1, -1, -1, -2, -1)-e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}\)
(-1, -1, -1, -1, -1, -1, -1, -1)-e_{1}-e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}\)
(0, 0, 0, 0, -2, -2, -2, -1)-2e_{5}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -2, -2, -1)-e_{4}-e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}s_{6}\)
(0, 0, -1, -1, -1, -1, -2, -1)-e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}\)
(0, -1, -1, -1, -1, -1, -1, -1)-e_{2}-e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}\)
(-1, -1, -1, -1, -1, -1, -1, 0)-e_{1}+e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -2, -2, -1)-e_{5}-e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{8}s_{7}s_{6}\)
(0, 0, 0, -1, -1, -1, -2, -1)-e_{4}-e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}\)
(0, 0, -1, -1, -1, -1, -1, -1)-e_{3}-e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}\)
(0, -1, -1, -1, -1, -1, -1, 0)-e_{2}+e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, -1, 0, 0)-e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -2, -2, -1)-2e_{6}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, -2, -1)-e_{5}-e_{7}\(s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{8}s_{7}\)
(0, 0, 0, -1, -1, -1, -1, -1)-e_{4}-e_{8}\(s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}\)
(0, 0, -1, -1, -1, -1, -1, 0)-e_{3}+e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, -1, 0, 0)-e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, 0, 0, 0)-e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, -2, -1)-e_{6}-e_{7}\(s_{7}s_{6}s_{8}s_{7}s_{6}s_{8}s_{7}\)
(0, 0, 0, 0, -1, -1, -1, -1)-e_{5}-e_{8}\(s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{8}\)
(0, 0, 0, -1, -1, -1, -1, 0)-e_{4}+e_{8}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, 0, 0)-e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, 0, 0, 0)-e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, -2, -1)-2e_{7}\(s_{7}s_{8}s_{7}\)
(0, 0, 0, 0, 0, -1, -1, -1)-e_{6}-e_{8}\(s_{6}s_{8}s_{7}s_{6}s_{8}\)
(0, 0, 0, 0, -1, -1, -1, 0)-e_{5}+e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, 0, 0)-e_{4}+e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, -1, -1)-e_{7}-e_{8}\(s_{8}s_{7}s_{8}\)
(0, 0, 0, 0, 0, -1, -1, 0)-e_{6}+e_{8}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, 0, 0)-e_{5}+e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, 0, -1)-2e_{8}\(s_{8}\)
(0, 0, 0, 0, 0, 0, -1, 0)-e_{7}+e_{8}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0, 0)-e_{6}+e_{7}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0, 0)e_{6}-e_{7}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 0)e_{7}-e_{8}\(s_{7}\)
(0, 0, 0, 0, 0, 0, 0, 1)2e_{8}\(s_{8}\)
(1, 1, 0, 0, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0, 0)e_{5}-e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 0)e_{6}-e_{8}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 1)e_{7}+e_{8}\(s_{8}s_{7}s_{8}\)
(1, 1, 1, 0, 0, 0, 0, 0)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 0, 0, 0, 0)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0, 0, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0, 0)e_{4}-e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 0)e_{5}-e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 1)e_{6}+e_{8}\(s_{6}s_{8}s_{7}s_{6}s_{8}\)
(0, 0, 0, 0, 0, 0, 2, 1)2e_{7}\(s_{7}s_{8}s_{7}\)
(1, 1, 1, 1, 0, 0, 0, 0)e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 0, 0, 0)e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 0, 0)e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 0)e_{4}-e_{8}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 1)e_{5}+e_{8}\(s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{8}\)
(0, 0, 0, 0, 0, 1, 2, 1)e_{6}+e_{7}\(s_{7}s_{6}s_{8}s_{7}s_{6}s_{8}s_{7}\)
(1, 1, 1, 1, 1, 0, 0, 0)e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 0, 0)e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 0)e_{3}-e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 1)e_{4}+e_{8}\(s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}\)
(0, 0, 0, 0, 1, 1, 2, 1)e_{5}+e_{7}\(s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{8}s_{7}\)
(0, 0, 0, 0, 0, 2, 2, 1)2e_{6}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 1, 0, 0)e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 0)e_{2}-e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 1)e_{3}+e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}\)
(0, 0, 0, 1, 1, 1, 2, 1)e_{4}+e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}\)
(0, 0, 0, 0, 1, 2, 2, 1)e_{5}+e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{8}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 1, 1, 0)e_{1}-e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 1)e_{2}+e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}\)
(0, 0, 1, 1, 1, 1, 2, 1)e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}\)
(0, 0, 0, 1, 1, 2, 2, 1)e_{4}+e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}s_{6}\)
(0, 0, 0, 0, 2, 2, 2, 1)2e_{5}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 1, 1)e_{1}+e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}\)
(0, 1, 1, 1, 1, 1, 2, 1)e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}\)
(0, 0, 1, 1, 1, 2, 2, 1)e_{3}+e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}s_{6}\)
(0, 0, 0, 1, 2, 2, 2, 1)e_{4}+e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 2, 1)e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}\)
(0, 1, 1, 1, 1, 2, 2, 1)e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}\)
(0, 0, 1, 1, 2, 2, 2, 1)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, 0, 2, 2, 2, 2, 1)2e_{4}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 1, 2, 2, 1)e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}\)
(0, 1, 1, 1, 2, 2, 2, 1)e_{2}+e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, 1, 2, 2, 2, 2, 1)e_{3}+e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 2, 2, 2, 1)e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}\)
(0, 1, 1, 2, 2, 2, 2, 1)e_{2}+e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 2, 2, 2, 2, 2, 1)2e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 2, 2, 1)e_{1}+e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 1, 2, 2, 2, 2, 2, 1)e_{2}+e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 2, 2, 2, 1)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 2, 2, 2, 2, 2, 2, 1)2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(1, 2, 2, 2, 2, 2, 2, 1)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(2, 2, 2, 2, 2, 2, 2, 1)2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-2, -2, -2, -2, -2, -2, -2, -1), (-1, -2, -2, -2, -2, -2, -2, -1), (0, -2, -2, -2, -2, -2, -2, -1), (-1, -1, -2, -2, -2, -2, -2, -1), (0, -1, -2, -2, -2, -2, -2, -1), (-1, -1, -1, -2, -2, -2, -2, -1), (0, 0, -2, -2, -2, -2, -2, -1), (0, -1, -1, -2, -2, -2, -2, -1), (-1, -1, -1, -1, -2, -2, -2, -1), (0, 0, -1, -2, -2, -2, -2, -1), (0, -1, -1, -1, -2, -2, -2, -1), (-1, -1, -1, -1, -1, -2, -2, -1), (0, 0, 0, -2, -2, -2, -2, -1), (0, 0, -1, -1, -2, -2, -2, -1), (0, -1, -1, -1, -1, -2, -2, -1), (-1, -1, -1, -1, -1, -1, -2, -1), (0, 0, 0, -1, -2, -2, -2, -1), (0, 0, -1, -1, -1, -2, -2, -1), (0, -1, -1, -1, -1, -1, -2, -1), (-1, -1, -1, -1, -1, -1, -1, -1), (0, 0, 0, 0, -2, -2, -2, -1), (0, 0, 0, -1, -1, -2, -2, -1), (0, 0, -1, -1, -1, -1, -2, -1), (0, -1, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, -1, 0), (0, 0, 0, 0, -1, -2, -2, -1), (0, 0, 0, -1, -1, -1, -2, -1), (0, 0, -1, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -1, -1, -1, 0, 0), (0, 0, 0, 0, 0, -2, -2, -1), (0, 0, 0, 0, -1, -1, -2, -1), (0, 0, 0, -1, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, 0, -1, -2, -1), (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, -1, -1, -1, 0, 0, 0), (-1, -1, -1, -1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, -2, -1), (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, -1, -1, 0, 0, 0, 0), (-1, -1, -1, 0, 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, -1, 0, 0, 0, 0, 0), (-1, -1, 0, 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, 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, 0, 0), (0, 0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 0, 1, 1), (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, 0, 0), (0, 0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 0, 0, 1, 1, 1), (0, 0, 0, 0, 0, 0, 2, 1), (1, 1, 1, 1, 0, 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, 0), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 0, 1, 2, 1), (1, 1, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 2, 1), (0, 0, 0, 0, 0, 2, 2, 1), (1, 1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1, 1, 1), (0, 0, 0, 1, 1, 1, 2, 1), (0, 0, 0, 0, 1, 2, 2, 1), (1, 1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1, 1, 1), (0, 0, 1, 1, 1, 1, 2, 1), (0, 0, 0, 1, 1, 2, 2, 1), (0, 0, 0, 0, 2, 2, 2, 1), (1, 1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 1, 1, 2, 1), (0, 0, 1, 1, 1, 2, 2, 1), (0, 0, 0, 1, 2, 2, 2, 1), (1, 1, 1, 1, 1, 1, 2, 1), (0, 1, 1, 1, 1, 2, 2, 1), (0, 0, 1, 1, 2, 2, 2, 1), (0, 0, 0, 2, 2, 2, 2, 1), (1, 1, 1, 1, 1, 2, 2, 1), (0, 1, 1, 1, 2, 2, 2, 1), (0, 0, 1, 2, 2, 2, 2, 1), (1, 1, 1, 1, 2, 2, 2, 1), (0, 1, 1, 2, 2, 2, 2, 1), (0, 0, 2, 2, 2, 2, 2, 1), (1, 1, 1, 2, 2, 2, 2, 1), (0, 1, 2, 2, 2, 2, 2, 1), (1, 1, 2, 2, 2, 2, 2, 1), (0, 2, 2, 2, 2, 2, 2, 1), (1, 2, 2, 2, 2, 2, 2, 1), (2, 2, 2, 2, 2, 2, 2, 1) The resulting Lie bracket pairing table follows.
Type C^{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_{-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}
(-2, -2, -2, -2, -2, -2, -2, -1)-2e_{1}g_{-64}0000000000000000000000000000000000000000000000000000000000000000g_{-64}0000000g_{-63}0000000g_{-61}000000g_{-59}000000g_{-56}00000g_{-53}00000g_{-49}0000g_{-45}0000-g_{-40}000-g_{-35}000-g_{-29}00-g_{-23}00-g_{-16}0-g_{-9}0-g_{-1}-h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}
(-1, -2, -2, -2, -2, -2, -2, -1)-e_{1}-e_{2}g_{-63}0000000000000000000000000000000000000000000000000000000000000002g_{-64}01/2g_{-63}0000002g_{-62}g_{-61}000000g_{-60}g_{-59}00000g_{-57}g_{-56}00000g_{-54}g_{-53}0000g_{-50}g_{-49}0000g_{-46}g_{-45}000g_{-41}-g_{-40}000-g_{-36}-g_{-35}00-g_{-30}-g_{-29}00-g_{-24}-g_{-23}0-g_{-17}-g_{-16}0-g_{-10}-g_{-9}-g_{-2}-g_{-1}-2h_{8}-4h_{7}-4h_{6}-4h_{5}-4h_{4}-4h_{3}-4h_{2}-2h_{1}-g_{1}
(0, -2, -2, -2, -2, -2, -2, -1)-2e_{2}g_{-62}000000000000000000000000000000000000000000000000000000000000000g_{-63}-g_{-62}g_{-62}0000000g_{-60}0000000g_{-57}000000g_{-54}000000g_{-50}00000g_{-46}00000g_{-41}0000-g_{-36}0000-g_{-30}000-g_{-24}000-g_{-17}00-g_{-10}00-g_{-2}0-h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-g_{1}0
(-1, -1, -2, -2, -2, -2, -2, -1)-e_{1}-e_{3}g_{-61}00000000000000000000000000000000000000000000000000000002g_{-64}000000g_{-63}01/2g_{-61}-1/2g_{-61}1/2g_{-61}00000g_{-60}0g_{-59}000002g_{-58}0g_{-56}0000g_{-55}0g_{-53}0000g_{-51}0g_{-49}000g_{-47}0g_{-45}000g_{-42}0-g_{-40}00g_{-37}0-g_{-35}00-g_{-31}0-g_{-29}0-g_{-25}0-g_{-23}0-g_{-18}0-g_{-16}-g_{-11}0-g_{-9}-g_{-3}-g_{-1}-2h_{8}-4h_{7}-4h_{6}-4h_{5}-4h_{4}-4h_{3}-2h_{2}-2h_{1}0-g_{2}-g_{9}
(0, -1, -2, -2, -2, -2, -2, -1)-e_{2}-e_{3}g_{-60}0000000000000000000000000000000000000000000000000000000g_{-63}0000002g_{-62}g_{-61}-1/2g_{-60}01/2g_{-60}0000002g_{-58}g_{-57}000000g_{-55}g_{-54}00000g_{-51}g_{-50}00000g_{-47}g_{-46}0000g_{-42}g_{-41}0000g_{-37}-g_{-36}000-g_{-31}-g_{-30}000-g_{-25}-g_{-24}00-g_{-18}-g_{-17}00-g_{-11}-g_{-10}0-g_{-3}-g_{-2}0-2h_{8}-4h_{7}-4h_{6}-4h_{5}-4h_{4}-4h_{3}-2h_{2}-g_{1}-g_{2}-g_{9}0
(-1, -1, -1, -2, -2, -2, -2, -1)-e_{1}-e_{4}g_{-59}0000000000000000000000000000000000000000000000002g_{-64}00000g_{-63}000000g_{-61}001/2g_{-59}0-1/2g_{-59}1/2g_{-59}0000g_{-57}00g_{-56}0000g_{-55}00g_{-53}0002g_{-52}00g_{-49}000g_{-48}00g_{-45}00g_{-43}00-g_{-40}00g_{-38}00-g_{-35}0g_{-32}00-g_{-29}0-g_{-26}00-g_{-23}-g_{-19}00-g_{-16}-g_{-12}0-g_{-9}-g_{-4}-g_{-1}0-2h_{8}-4h_{7}-4h_{6}-4h_{5}-4h_{4}-2h_{3}-2h_{2}-2h_{1}0-g_{3}0-g_{10}-g_{16}
(0, 0, -2, -2, -2, -2, -2, -1)-2e_{3}g_{-58}0000000000000000000000000000000000000000000000000000000g_{-61}000000g_{-60}00-g_{-58}g_{-58}0000000g_{-55}0000000g_{-51}000000g_{-47}000000g_{-42}00000g_{-37}00000-g_{-31}0000-g_{-25}0000-g_{-18}000-g_{-11}000-g_{-3}00-h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}0-g_{2}-g_{9}000
(0, -1, -1, -2, -2, -2, -2, -1)-e_{2}-e_{4}g_{-57}000000000000000000000000000000000000000000000000g_{-63}000002g_{-62}000000g_{-60}0g_{-59}-1/2g_{-57}1/2g_{-57}-1/2g_{-57}1/2g_{-57}00000g_{-55}0g_{-54}000002g_{-52}0g_{-50}0000g_{-48}0g_{-46}0000g_{-43}0g_{-41}000g_{-38}0-g_{-36}000g_{-32}0-g_{-30}00-g_{-26}0-g_{-24}00-g_{-19}0-g_{-17}0-g_{-12}0-g_{-10}0-g_{-4}-g_{-2}0-2h_{8}-4h_{7}-4h_{6}-4h_{5}-4h_{4}-2h_{3}-2h_{2}0-g_{1}-g_{3}0-g_{10}-g_{16}0
(-1, -1, -1, -1, -2, -2, -2, -1)-e_{1}-e_{5}g_{-56}000000000000000000000000000000000000000002g_{-64}00000g_{-63}00000g_{-61}000000g_{-59}0001/2g_{-56}00-1/2g_{-56}1/2g_{-56}000g_{-54}000g_{-53}000g_{-51}000g_{-49}00g_{-48}000g_{-45}002g_{-44}000-g_{-40}0g_{-39}000-g_{-35}0g_{-33}000-g_{-29}g_{-27}000-g_{-23}-g_{-20}00-g_{-16}-g_{-13}0-g_{-9}0-g_{-5}-g_{-1}0-2h_{8}-4h_{7}-4h_{6}-4h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}00-g_{4}0-g_{11}0-g_{17}-g_{23}
(0, 0, -1, -2, -2, -2, -2, -1)-e_{3}-e_{4}g_{-55}000000000000000000000000000000000000000000000000g_{-61}00000g_{-60}g_{-59}000002g_{-58}g_{-57}00-1/2g_{-55}01/2g_{-55}0000002g_{-52}g_{-51}000000g_{-48}g_{-47}00000g_{-43}g_{-42}00000g_{-38}g_{-37}0000g_{-32}-g_{-31}0000-g_{-26}-g_{-25}000-g_{-19}-g_{-18}000-g_{-12}-g_{-11}00-g_{-4}-g_{-3}00-2h_{8}-4h_{7}-4h_{6}-4h_{5}-4h_{4}-2h_{3}0-g_{2}-g_{3}-g_{9}-g_{10}-g_{16}000
(0, -1, -1, -1, -2, -2, -2, -1)-e_{2}-e_{5}g_{-54}00000000000000000000000000000000000000000g_{-63}000002g_{-62}00000g_{-60}000000g_{-57}00g_{-56}-1/2g_{-54}1/2g_{-54}0-1/2g_{-54}1/2g_{-54}0000g_{-51}00g_{-50}0000g_{-48}00g_{-46}0002g_{-44}00g_{-41}000g_{-39}00-g_{-36}00g_{-33}00-g_{-30}00g_{-27}00-g_{-24}0-g_{-20}00-g_{-17}0-g_{-13}0-g_{-10}0-g_{-5}-g_{-2}00-2h_{8}-4h_{7}-4h_{6}-4h_{5}-2h_{4}-2h_{3}-2h_{2}0-g_{1}-g_{4}00-g_{11}0-g_{17}-g_{23}0
(-1, -1, -1, -1, -1, -2, -2, -1)-e_{1}-e_{6}g_{-53}000000000000000000000000000000000002g_{-64}0000g_{-63}00000g_{-61}00000g_{-59}000000g_{-56}00001/2g_{-53}000-1/2g_{-53}1/2g_{-53}00g_{-50}0000g_{-49}00g_{-47}0000g_{-45}0g_{-43}0000-g_{-40}0g_{-39}0000-g_{-35}2g_{-34}0000-g_{-29}g_{-28}000-g_{-23}g_{-21}00-g_{-16}0-g_{-14}0-g_{-9}0-g_{-6}-g_{-1}00-2h_{8}-4h_{7}-4h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-2h_{1}00-g_{5}00-g_{12}0-g_{18}0-g_{24}-g_{29}
(0, 0, 0, -2, -2, -2, -2, -1)-2e_{4}g_{-52}000000000000000000000000000000000000000000000000g_{-59}00000g_{-57}000000g_{-55}0000-g_{-52}g_{-52}0000000g_{-48}0000000g_{-43}000000g_{-38}000000g_{-32}00000-g_{-26}00000-g_{-19}0000-g_{-12}0000-g_{-4}000-h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}00-g_{3}0-g_{10}0-g_{16}00000
(0, 0, -1, -1, -2, -2, -2, -1)-e_{3}-e_{5}g_{-51}00000000000000000000000000000000000000000g_{-61}00000g_{-60}000002g_{-58}0g_{-56}0000g_{-55}0g_{-54}00-1/2g_{-51}1/2g_{-51}-1/2g_{-51}1/2g_{-51}00000g_{-48}0g_{-47}000002g_{-44}0g_{-42}0000g_{-39}0g_{-37}0000g_{-33}0-g_{-31}000g_{-27}0-g_{-25}000-g_{-20}0-g_{-18}00-g_{-13}0-g_{-11}00-g_{-5}-g_{-3}00-2h_{8}-4h_{7}-4h_{6}-4h_{5}-2h_{4}-2h_{3}00-g_{2}-g_{4}-g_{9}0-g_{11}0-g_{17}-g_{23}000
(0, -1, -1, -1, -1, -2, -2, -1)-e_{2}-e_{6}g_{-50}00000000000000000000000000000000000g_{-63}00002g_{-62}00000g_{-60}00000g_{-57}000000g_{-54}000g_{-53}-1/2g_{-50}1/2g_{-50}00-1/2g_{-50}1/2g_{-50}000g_{-47}000g_{-46}000g_{-43}000g_{-41}00g_{-39}000-g_{-36}002g_{-34}000-g_{-30}0g_{-28}000-g_{-24}0g_{-21}00-g_{-17}0-g_{-14}0-g_{-10}00-g_{-6}-g_{-2}00-2h_{8}-4h_{7}-4h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}00-g_{1}-g_{5}00-g_{12}00-g_{18}0-g_{24}-g_{29}0
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(0, 0, 0, 0, 2, 2, 2, 1)2e_{5}g_{44}00000000g_{-23}0g_{-17}00g_{-11}00g_{-4}000h_{8}+2h_{7}+2h_{6}+2h_{5}0000g_{5}00000g_{13}00000g_{20}000000-g_{27}000000-g_{33}0000000-g_{39}0000000g_{44}-g_{44}000000-g_{48}000000-g_{51}00000-g_{54}00000-g_{56}00000000000000000000000000000000000000000
(1, 1, 1, 1, 1, 1, 1, 1)e_{1}+e_{8}g_{45}g_{-40}g_{-36}0g_{-31}0g_{-26}00g_{-20}00g_{-14}000g_{-7}0002h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}000g_{1}-2g_{8}00g_{9}0-g_{15}00g_{16}00-g_{21}0g_{23}000-g_{27}0g_{29}0000-g_{32}g_{35}00000-g_{37}g_{40}000000-g_{41}-1/2g_{45}000001/2g_{45}-g_{45}000000-g_{49}000000-g_{53}00000-g_{56}00000-g_{59}0000-g_{61}0000-g_{63}000-2g_{64}000000000000000000000000
(0, 1, 1, 1, 1, 1, 2, 1)e_{2}+e_{7}g_{46}0g_{-35}g_{-30}0g_{-25}00g_{-19}00g_{-13}000g_{-6}g_{-1}002h_{8}+4h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}000g_{2}g_{7}00g_{10}0-g_{15}00g_{17}00-2g_{22}0g_{24}000-g_{28}0g_{30}0000-g_{33}0g_{36}0000-g_{38}00-g_{41}0000-g_{42}01/2g_{46}-1/2g_{46}0001/2g_{46}-1/2g_{46}0-g_{49}0000-g_{50}000000-g_{54}00000-g_{57}00000-g_{60}0000-2g_{62}0000-g_{63}00000000000000000000000000000
(0, 0, 1, 1, 1, 2, 2, 1)e_{3}+e_{6}g_{47}000g_{-29}g_{-24}0g_{-18}00g_{-12}0g_{-9}0g_{-5}g_{-2}002h_{8}+4h_{7}+4h_{6}+2h_{5}+2h_{4}+2h_{3}000g_{3}g_{6}00g_{11}0g_{14}00g_{18}00-g_{21}00g_{25}00-g_{28}000g_{31}00-2g_{34}000-g_{37}00-g_{39}0000-g_{42}00-g_{43}0001/2g_{47}-1/2g_{47}01/2g_{47}-1/2g_{47}000-g_{50}00-g_{51}000-g_{53}00-g_{55}00000-2g_{58}00000-g_{60}0000-g_{61}00000000000000000000000000000000000
(0, 0, 0, 1, 2, 2, 2, 1)e_{4}+e_{5}g_{48}00000g_{-23}0g_{-17}g_{-16}g_{-11}g_{-10}0g_{-4}g_{-3}002h_{8}+4h_{7}+4h_{6}+4h_{5}+2h_{4}000g_{4}g_{5}000g_{12}g_{13}0000g_{19}g_{20}0000g_{26}-g_{27}00000-g_{32}-g_{33}00000-g_{38}-g_{39}000000-g_{43}-2g_{44}000001/2g_{48}0-1/2g_{48}00000-g_{51}-2g_{52}00000-g_{54}-g_{55}0000-g_{56}-g_{57}00000-g_{59}00000000000000000000000000000000000000000
(1, 1, 1, 1, 1, 1, 2, 1)e_{1}+e_{7}g_{49}g_{-35}g_{-30}0g_{-25}0g_{-19}00g_{-13}00g_{-6}0002h_{8}+4h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}00g_{1}g_{7}00g_{9}0-g_{15}0g_{16}00-2g_{22}0g_{23}000-g_{28}g_{29}0000-g_{33}g_{35}00000-g_{38}g_{40}00000-g_{42}0-g_{45}00000-g_{46}-1/2g_{49}00001/2g_{49}-1/2g_{49}000000-g_{53}000000-g_{56}00000-g_{59}00000-g_{61}0000-g_{63}0000-2g_{64}00000000000000000000000000000
(0, 1, 1, 1, 1, 2, 2, 1)e_{2}+e_{6}g_{50}0g_{-29}g_{-24}0g_{-18}00g_{-12}00g_{-5}g_{-1}002h_{8}+4h_{7}+4h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}00g_{2}g_{6}00g_{10}0g_{14}0g_{17}00-g_{21}0g_{24}000-g_{28}0g_{30}000-2g_{34}00g_{36}000-g_{39}00-g_{41}000-g_{43}000-g_{46}000-g_{47}01/2g_{50}-1/2g_{50}001/2g_{50}-1/2g_{50}00-g_{53}000-g_{54}000000-g_{57}00000-g_{60}00000-2g_{62}0000-g_{63}00000000000000000000000000000000000
(0, 0, 1, 1, 2, 2, 2, 1)e_{3}+e_{5}g_{51}000g_{-23}g_{-17}0g_{-11}0g_{-9}g_{-4}g_{-2}002h_{8}+4h_{7}+4h_{6}+4h_{5}+2h_{4}+2h_{3}00g_{3}g_{5}00g_{11}0g_{13}00g_{18}0g_{20}000g_{25}0-g_{27}000g_{31}0-g_{33}0000-g_{37}0-g_{39}0000-g_{42}0-2g_{44}00000-g_{47}0-g_{48}0001/2g_{51}-1/2g_{51}1/2g_{51}-1/2g_{51}0000-g_{54}0-g_{55}0000-g_{56}0-2g_{58}00000-g_{60}00000-g_{61}00000000000000000000000000000000000000000
(0, 0, 0, 2, 2, 2, 2, 1)2e_{4}g_{52}00000g_{-16}0g_{-10}0g_{-3}00h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}000g_{4}0000g_{12}0000g_{19}00000g_{26}00000-g_{32}000000-g_{38}000000-g_{43}0000000-g_{48}00000g_{52}-g_{52}000000-g_{55}000000-g_{57}00000-g_{59}000000000000000000000000000000000000000000000000
(1, 1, 1, 1, 1, 2, 2, 1)e_{1}+e_{6}g_{53}g_{-29}g_{-24}0g_{-18}0g_{-12}00g_{-5}002h_{8}+4h_{7}+4h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}00g_{1}g_{6}0g_{9}0g_{14}0g_{16}00-g_{21}g_{23}000-g_{28}g_{29}0000-2g_{34}g_{35}0000-g_{39}0g_{40}0000-g_{43}0-g_{45}0000-g_{47}00-g_{49}0000-g_{50}-1/2g_{53}0001/2g_{53}-1/2g_{53}000000-g_{56}000000-g_{59}00000-g_{61}00000-g_{63}0000-2g_{64}00000000000000000000000000000000000
(0, 1, 1, 1, 2, 2, 2, 1)e_{2}+e_{5}g_{54}0g_{-23}g_{-17}0g_{-11}00g_{-4}g_{-1}02h_{8}+4h_{7}+4h_{6}+4h_{5}+2h_{4}+2h_{3}+2h_{2}00g_{2}g_{5}0g_{10}0g_{13}0g_{17}00g_{20}0g_{24}00-g_{27}00g_{30}00-g_{33}00g_{36}00-g_{39}000-g_{41}00-2g_{44}000-g_{46}00-g_{48}0000-g_{50}00-g_{51}01/2g_{54}-1/2g_{54}01/2g_{54}-1/2g_{54}000-g_{56}00-g_{57}000000-g_{60}00000-2g_{62}00000-g_{63}00000000000000000000000000000000000000000
(0, 0, 1, 2, 2, 2, 2, 1)e_{3}+e_{4}g_{55}000g_{-16}g_{-10}g_{-9}g_{-3}g_{-2}02h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+2h_{3}00g_{3}g_{4}00g_{11}g_{12}000g_{18}g_{19}000g_{25}g_{26}0000g_{31}-g_{32}0000-g_{37}-g_{38}00000-g_{42}-g_{43}00000-g_{47}-g_{48}000000-g_{51}-2g_{52}0001/2g_{55}0-1/2g_{55}00000-g_{57}-2g_{58}00000-g_{59}-g_{60}00000-g_{61}000000000000000000000000000000000000000000000000
(1, 1, 1, 1, 2, 2, 2, 1)e_{1}+e_{5}g_{56}g_{-23}g_{-17}0g_{-11}0g_{-4}002h_{8}+4h_{7}+4h_{6}+4h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}g_{5}0g_{9}0g_{13}g_{16}00g_{20}g_{23}000-g_{27}g_{29}000-g_{33}0g_{35}000-g_{39}0g_{40}000-2g_{44}00-g_{45}000-g_{48}00-g_{49}000-g_{51}000-g_{53}000-g_{54}-1/2g_{56}001/2g_{56}-1/2g_{56}000000-g_{59}000000-g_{61}00000-g_{63}00000-2g_{64}00000000000000000000000000000000000000000
(0, 1, 1, 2, 2, 2, 2, 1)e_{2}+e_{4}g_{57}0g_{-16}g_{-10}0g_{-3}g_{-1}02h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+2h_{3}+2h_{2}0g_{2}g_{4}0g_{10}0g_{12}0g_{17}0g_{19}00g_{24}0g_{26}00g_{30}0-g_{32}000g_{36}0-g_{38}000-g_{41}0-g_{43}0000-g_{46}0-g_{48}0000-g_{50}0-2g_{52}00000-g_{54}0-g_{55}01/2g_{57}-1/2g_{57}1/2g_{57}-1/2g_{57}0000-g_{59}0-g_{60}000000-2g_{62}00000-g_{63}000000000000000000000000000000000000000000000000
(0, 0, 2, 2, 2, 2, 2, 1)2e_{3}g_{58}000g_{-9}g_{-2}0h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}00g_{3}000g_{11}000g_{18}0000g_{25}0000g_{31}00000-g_{37}00000-g_{42}000000-g_{47}000000-g_{51}0000000-g_{55}000g_{58}-g_{58}000000-g_{60}000000-g_{61}0000000000000000000000000000000000000000000000000000000
(1, 1, 1, 2, 2, 2, 2, 1)e_{1}+e_{4}g_{59}g_{-16}g_{-10}0g_{-3}02h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}g_{4}g_{9}0g_{12}g_{16}00g_{19}g_{23}00g_{26}0g_{29}00-g_{32}0g_{35}00-g_{38}00g_{40}00-g_{43}00-g_{45}00-g_{48}000-g_{49}00-2g_{52}000-g_{53}00-g_{55}0000-g_{56}00-g_{57}-1/2g_{59}01/2g_{59}-1/2g_{59}000000-g_{61}000000-g_{63}00000-2g_{64}000000000000000000000000000000000000000000000000
(0, 1, 2, 2, 2, 2, 2, 1)e_{2}+e_{3}g_{60}0g_{-9}g_{-2}g_{-1}2h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+2h_{2}0g_{2}g_{3}0g_{10}g_{11}00g_{17}g_{18}00g_{24}g_{25}000g_{30}g_{31}000g_{36}-g_{37}0000-g_{41}-g_{42}0000-g_{46}-g_{47}00000-g_{50}-g_{51}00000-g_{54}-g_{55}000000-g_{57}-2g_{58}01/2g_{60}0-1/2g_{60}00000-g_{61}-2g_{62}000000-g_{63}0000000000000000000000000000000000000000000000000000000
(1, 1, 2, 2, 2, 2, 2, 1)e_{1}+e_{3}g_{61}g_{-9}g_{-2}02h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+2h_{2}+2h_{1}g_{1}g_{3}g_{9}0g_{11}g_{16}0g_{18}0g_{23}0g_{25}0g_{29}0g_{31}00g_{35}0-g_{37}00g_{40}0-g_{42}000-g_{45}0-g_{47}000-g_{49}0-g_{51}0000-g_{53}0-g_{55}0000-g_{56}0-2g_{58}00000-g_{59}0-g_{60}-1/2g_{61}1/2g_{61}-1/2g_{61}000000-g_{63}000000-2g_{64}0000000000000000000000000000000000000000000000000000000
(0, 2, 2, 2, 2, 2, 2, 1)2e_{2}g_{62}0g_{-1}h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}0g_{2}00g_{10}00g_{17}000g_{24}000g_{30}0000g_{36}0000-g_{41}00000-g_{46}00000-g_{50}000000-g_{54}000000-g_{57}0000000-g_{60}0g_{62}-g_{62}000000-g_{63}000000000000000000000000000000000000000000000000000000000000000
(1, 2, 2, 2, 2, 2, 2, 1)e_{1}+e_{2}g_{63}g_{-1}2h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}g_{1}g_{2}g_{9}g_{10}0g_{16}g_{17}0g_{23}g_{24}00g_{29}g_{30}00g_{35}g_{36}000g_{40}-g_{41}000-g_{45}-g_{46}0000-g_{49}-g_{50}0000-g_{53}-g_{54}00000-g_{56}-g_{57}00000-g_{59}-g_{60}000000-g_{61}-2g_{62}0-1/2g_{63}000000-2g_{64}000000000000000000000000000000000000000000000000000000000000000
(2, 2, 2, 2, 2, 2, 2, 1)2e_{1}g_{64}h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}g_{1}0g_{9}0g_{16}00g_{23}00g_{29}000g_{35}000g_{40}0000-g_{45}0000-g_{49}00000-g_{53}00000-g_{56}000000-g_{59}000000-g_{61}0000000-g_{63}-g_{64}00000000000000000000000000000000000000000000000000000000000000000000000
We define the symmetric Cartan matrix
by requesting that the entry in the i-th row and j-th column
be the scalar product of the i^th and j^th roots. The symmetric Cartan matrix is:
\(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0\\ 0 & -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & -1/2 & 1 & -1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & -1/2 & 1 & -1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1 & -1/2 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/2 & 1 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 4 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 4 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 4 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 4 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 4 & -2\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1/64
Half sum of positive roots: (8, 15, 21, 26, 30, 33, 35, 18)= \(\displaystyle 8\varepsilon_{1}+7\varepsilon_{2}+6\varepsilon_{3}+5\varepsilon_{4}+4\varepsilon_{5}+3\varepsilon_{6}+2\varepsilon_{7}+\varepsilon_{8}\)
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):
(1, 1, 1, 1, 1, 1, 1, 1/2) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 2, 2, 2, 1) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3, 3, 3, 3/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 4, 4, 4, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5, 5, 5, 5/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
(1, 2, 3, 4, 5, 6, 6, 3) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}\)
(1, 2, 3, 4, 5, 6, 7, 7/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}\)
(1, 2, 3, 4, 5, 6, 7, 4) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}+\varepsilon_{8}\)
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) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)
(0, 0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{7}-\varepsilon_{8}\)
(0, 0, 0, 0, 0, 0, 0, 1) = \(\displaystyle 2\varepsilon_{8}\)