so(17), type \(B^{1}_8\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: B^{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
(-1, -2, -2, -2, -2, -2, -2, -2)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -2, -2, -2, -2)-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, -2)-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, -2)-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, -1, -2, -2, -2, -2, -2)-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, -2)-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, -2)-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, -2)-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, -2)-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, -1, -1, -2, -2, -2, -2)-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, -2)-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, -2)-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, -2)-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, -2)-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, -2)-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, -2)-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, -1, -1, -2, -2, -2)-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, -2)-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, -2)-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, -1)-e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -2, -2, -2)-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, -2)-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, -2)-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, -1)-e_{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, -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, -1, -2, -2)-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, -2)-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, -1)-e_{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, -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, -1, -2, -2)-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, -2)-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, -1)-e_{4}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, -1, 0)-e_{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, -1, -2)-e_{6}-e_{8}\(s_{6}s_{8}s_{7}s_{6}s_{8}\)
(0, 0, 0, 0, -1, -1, -1, -1)-e_{5}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(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, -1, -2)-e_{7}-e_{8}\(s_{8}s_{7}s_{8}\)
(0, 0, 0, 0, 0, -1, -1, -1)-e_{6}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(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}\(s_{7}s_{8}s_{7}\)
(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)-e_{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)e_{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}\(s_{7}s_{8}s_{7}\)
(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}\(s_{6}s_{7}s_{8}s_{7}s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 2)e_{7}+e_{8}\(s_{8}s_{7}s_{8}\)
(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}\(s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 2)e_{6}+e_{8}\(s_{6}s_{8}s_{7}s_{6}s_{8}\)
(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}\(s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 2)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, 2)e_{6}+e_{7}\(s_{7}s_{6}s_{8}s_{7}s_{6}s_{8}s_{7}\)
(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}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 2)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, 2)e_{5}+e_{7}\(s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{8}s_{7}\)
(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}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 2)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, 2)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, 2)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, 1)e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 2)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, 2)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, 2)e_{4}+e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 1, 1, 2)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, 2)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, 2)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, 2)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, 2)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, 2)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, 2)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{8}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 2, 2, 2)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, 2)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, 2)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, 2)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, 2)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, 2, 2, 2, 2, 2)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, 2)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, 2)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 2, 2, 2, 2, 2, 2, 2)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
Comma delimited list of roots: (-1, -2, -2, -2, -2, -2, -2, -2), (-1, -1, -2, -2, -2, -2, -2, -2), (0, -1, -2, -2, -2, -2, -2, -2), (-1, -1, -1, -2, -2, -2, -2, -2), (0, -1, -1, -2, -2, -2, -2, -2), (-1, -1, -1, -1, -2, -2, -2, -2), (0, 0, -1, -2, -2, -2, -2, -2), (0, -1, -1, -1, -2, -2, -2, -2), (-1, -1, -1, -1, -1, -2, -2, -2), (0, 0, -1, -1, -2, -2, -2, -2), (0, -1, -1, -1, -1, -2, -2, -2), (-1, -1, -1, -1, -1, -1, -2, -2), (0, 0, 0, -1, -2, -2, -2, -2), (0, 0, -1, -1, -1, -2, -2, -2), (0, -1, -1, -1, -1, -1, -2, -2), (-1, -1, -1, -1, -1, -1, -1, -2), (0, 0, 0, -1, -1, -2, -2, -2), (0, 0, -1, -1, -1, -1, -2, -2), (0, -1, -1, -1, -1, -1, -1, -2), (-1, -1, -1, -1, -1, -1, -1, -1), (0, 0, 0, 0, -1, -2, -2, -2), (0, 0, 0, -1, -1, -1, -2, -2), (0, 0, -1, -1, -1, -1, -1, -2), (0, -1, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, -1, 0), (0, 0, 0, 0, -1, -1, -2, -2), (0, 0, 0, -1, -1, -1, -1, -2), (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, -1, -2, -2), (0, 0, 0, 0, -1, -1, -1, -2), (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, -1, -2), (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, -1, -2), (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, 1, 2), (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, 1, 2), (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, 1, 2), (0, 0, 0, 0, 0, 1, 2, 2), (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, 1, 2), (0, 0, 0, 0, 1, 1, 2, 2), (1, 1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1, 1, 1), (0, 0, 1, 1, 1, 1, 1, 2), (0, 0, 0, 1, 1, 1, 2, 2), (0, 0, 0, 0, 1, 2, 2, 2), (1, 1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 1, 1, 1, 2), (0, 0, 1, 1, 1, 1, 2, 2), (0, 0, 0, 1, 1, 2, 2, 2), (1, 1, 1, 1, 1, 1, 1, 2), (0, 1, 1, 1, 1, 1, 2, 2), (0, 0, 1, 1, 1, 2, 2, 2), (0, 0, 0, 1, 2, 2, 2, 2), (1, 1, 1, 1, 1, 1, 2, 2), (0, 1, 1, 1, 1, 2, 2, 2), (0, 0, 1, 1, 2, 2, 2, 2), (1, 1, 1, 1, 1, 2, 2, 2), (0, 1, 1, 1, 2, 2, 2, 2), (0, 0, 1, 2, 2, 2, 2, 2), (1, 1, 1, 1, 2, 2, 2, 2), (0, 1, 1, 2, 2, 2, 2, 2), (1, 1, 1, 2, 2, 2, 2, 2), (0, 1, 2, 2, 2, 2, 2, 2), (1, 1, 2, 2, 2, 2, 2, 2), (1, 2, 2, 2, 2, 2, 2, 2) The resulting Lie bracket pairing table follows.
Type B^{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}
(-1, -2, -2, -2, -2, -2, -2, -2)-e_{1}-e_{2}g_{-64}00000000000000000000000000000000000000000000000000000000000000000g_{-64}0000000g_{-63}000000-g_{-62}g_{-61}00000-g_{-60}g_{-59}00000-g_{-57}g_{-56}0000-g_{-54}g_{-53}0000-g_{-50}g_{-49}000-g_{-46}g_{-45}000-g_{-41}g_{-40}00-g_{-36}g_{-35}00-g_{-30}g_{-29}0-g_{-24}g_{-23}0-g_{-17}g_{-16}-g_{-10}g_{-9}-g_{-2}-2h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-h_{1}
(-1, -1, -2, -2, -2, -2, -2, -2)-e_{1}-e_{3}g_{-63}00000000000000000000000000000000000000000000000000000000000000g_{-64}0g_{-63}-g_{-63}g_{-63}00000g_{-62}0g_{-61}0000000g_{-59}0000-g_{-58}0g_{-56}0000-g_{-55}0g_{-53}000-g_{-51}0g_{-49}000-g_{-47}0g_{-45}00-g_{-42}0g_{-40}00-g_{-37}0g_{-35}0-g_{-31}0g_{-29}0-g_{-25}0g_{-23}-g_{-18}0g_{-16}-g_{-11}0-g_{-3}-g_{-1}-2h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-h_{2}-h_{1}-g_{2}
(0, -1, -2, -2, -2, -2, -2, -2)-e_{2}-e_{3}g_{-62}0000000000000000000000000000000000000000000000000000000-g_{-64}0000000g_{-63}-g_{-62}0g_{-62}0000000g_{-60}000000-g_{-58}g_{-57}00000-g_{-55}g_{-54}00000-g_{-51}g_{-50}0000-g_{-47}g_{-46}0000-g_{-42}g_{-41}000-g_{-37}g_{-36}000-g_{-31}g_{-30}00-g_{-25}g_{-24}00-g_{-18}g_{-17}0-g_{-11}g_{-10}0-g_{-3}0-2h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-h_{2}-g_{1}g_{9}
(-1, -1, -1, -2, -2, -2, -2, -2)-e_{1}-e_{4}g_{-61}000000000000000000000000000000000000000000000000000000g_{-64}000000g_{-63}00g_{-61}0-g_{-61}g_{-61}0000g_{-60}00g_{-59}0000g_{-58}00g_{-56}000000g_{-53}000-g_{-52}00g_{-49}00-g_{-48}00g_{-45}00-g_{-43}00g_{-40}0-g_{-38}00g_{-35}0-g_{-32}00g_{-29}-g_{-26}00g_{-23}-g_{-19}00-g_{-12}0-g_{-9}-g_{-4}-g_{-1}-2h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}-h_{1}0-g_{3}-g_{10}
(0, -1, -1, -2, -2, -2, -2, -2)-e_{2}-e_{4}g_{-60}000000000000000000000000000000000000000000000000-g_{-64}000000000000g_{-62}0g_{-61}-g_{-60}g_{-60}-g_{-60}g_{-60}00000g_{-58}0g_{-57}0000000g_{-54}0000-g_{-52}0g_{-50}0000-g_{-48}0g_{-46}000-g_{-43}0g_{-41}000-g_{-38}0g_{-36}00-g_{-32}0g_{-30}00-g_{-26}0g_{-24}0-g_{-19}0g_{-17}0-g_{-12}00-g_{-4}-g_{-2}0-2h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}-g_{1}-g_{3}0g_{16}
(-1, -1, -1, -1, -2, -2, -2, -2)-e_{1}-e_{5}g_{-59}00000000000000000000000000000000000000000000000g_{-64}00000g_{-63}000000g_{-61}000g_{-59}00-g_{-59}g_{-59}000g_{-57}000g_{-56}000g_{-55}000g_{-53}00g_{-52}000g_{-49}000000g_{-45}0-g_{-44}000g_{-40}0-g_{-39}000g_{-35}-g_{-33}000g_{-29}-g_{-27}000-g_{-20}00-g_{-16}-g_{-13}0-g_{-9}-g_{-5}-g_{-1}0-2h_{8}-2h_{7}-2h_{6}-2h_{5}-h_{4}-h_{3}-h_{2}-h_{1}0-g_{4}0-g_{11}-g_{17}
(0, 0, -1, -2, -2, -2, -2, -2)-e_{3}-e_{4}g_{-58}000000000000000000000000000000000000000000000000-g_{-63}00000-g_{-62}g_{-61}000000g_{-60}00-g_{-58}0g_{-58}0000000g_{-55}000000-g_{-52}g_{-51}00000-g_{-48}g_{-47}00000-g_{-43}g_{-42}0000-g_{-38}g_{-37}0000-g_{-32}g_{-31}000-g_{-26}g_{-25}000-g_{-19}g_{-18}00-g_{-12}g_{-11}00-g_{-4}00-2h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-h_{3}0-g_{2}-g_{9}g_{10}g_{16}0
(0, -1, -1, -1, -2, -2, -2, -2)-e_{2}-e_{5}g_{-57}00000000000000000000000000000000000000000-g_{-64}00000000000g_{-62}000000g_{-60}00g_{-59}-g_{-57}g_{-57}0-g_{-57}g_{-57}0000g_{-55}00g_{-54}0000g_{-52}00g_{-50}000000g_{-46}000-g_{-44}00g_{-41}00-g_{-39}00g_{-36}00-g_{-33}00g_{-30}0-g_{-27}00g_{-24}0-g_{-20}000-g_{-13}0-g_{-10}0-g_{-5}-g_{-2}0-2h_{8}-2h_{7}-2h_{6}-2h_{5}-h_{4}-h_{3}-h_{2}0-g_{1}-g_{4}0-g_{11}0g_{23}
(-1, -1, -1, -1, -1, -2, -2, -2)-e_{1}-e_{6}g_{-56}0000000000000000000000000000000000000000g_{-64}00000g_{-63}00000g_{-61}000000g_{-59}0000g_{-56}000-g_{-56}g_{-56}00g_{-54}0000g_{-53}00g_{-51}0000g_{-49}0g_{-48}0000g_{-45}0g_{-44}0000g_{-40}00000g_{-35}-g_{-34}0000-g_{-28}000-g_{-23}-g_{-21}00-g_{-16}-g_{-14}0-g_{-9}0-g_{-6}-g_{-1}0-2h_{8}-2h_{7}-2h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}00-g_{5}0-g_{12}0-g_{18}-g_{24}
(0, 0, -1, -1, -2, -2, -2, -2)-e_{3}-e_{5}g_{-55}00000000000000000000000000000000000000000-g_{-63}00000-g_{-62}0000000g_{-59}0000g_{-58}0g_{-57}00-g_{-55}g_{-55}-g_{-55}g_{-55}00000g_{-52}0g_{-51}0000000g_{-47}0000-g_{-44}0g_{-42}0000-g_{-39}0g_{-37}000-g_{-33}0g_{-31}000-g_{-27}0g_{-25}00-g_{-20}0g_{-18}00-g_{-13}000-g_{-5}-g_{-3}00-2h_{8}-2h_{7}-2h_{6}-2h_{5}-h_{4}-h_{3}0-g_{2}-g_{4}-g_{9}00g_{17}g_{23}0
(0, -1, -1, -1, -1, -2, -2, -2)-e_{2}-e_{6}g_{-54}00000000000000000000000000000000000-g_{-64}0000000000g_{-62}00000g_{-60}000000g_{-57}000g_{-56}-g_{-54}g_{-54}00-g_{-54}g_{-54}000g_{-51}000g_{-50}000g_{-48}000g_{-46}00g_{-44}000g_{-41}000000g_{-36}0-g_{-34}000g_{-30}0-g_{-28}0000-g_{-21}00-g_{-17}0-g_{-14}0-g_{-10}0-g_{-6}-g_{-2}00-2h_{8}-2h_{7}-2h_{6}-h_{5}-h_{4}-h_{3}-h_{2}0-g_{1}-g_{5}00-g_{12}0-g_{18}0g_{29}
(-1, -1, -1, -1, -1, -1, -2, -2)-e_{1}-e_{7}g_{-53}0000000000000000000000000000000000g_{-64}0000g_{-63}00000g_{-61}00000g_{-59}000000g_{-56}00000g_{-53}0000-g_{-53}g_{-53}0g_{-50}00000g_{-49}0g_{-47}00000g_{-45}g_{-43}00000g_{-40}g_{-39}00000g_{-34}0000-g_{-29}0000-g_{-23}-g_{-22}00-g_{-16}0-g_{-15}0-g_{-9}0-g_{-7}-g_{-1}00-2h_{8}-2h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}00-g_{6}00-g_{13}0-g_{19}0-g_{25}-g_{30}
(0, 0, 0, -1, -2, -2, -2, -2)-e_{4}-e_{5}g_{-52}00000000000000000000000000000000000000000-g_{-61}00000-g_{-60}g_{-59}0000-g_{-58}g_{-57}000000g_{-55}0000-g_{-52}0g_{-52}0000000g_{-48}000000-g_{-44}g_{-43}00000-g_{-39}g_{-38}00000-g_{-33}g_{-32}0000-g_{-27}g_{-26}0000-g_{-20}g_{-19}000-g_{-13}g_{-12}000-g_{-5}000-2h_{8}-2h_{7}-2h_{6}-2h_{5}-h_{4}00-g_{3}0-g_{10}g_{11}-g_{16}g_{17}g_{23}000
(0, 0, -1, -1, -1, -2, -2, -2)-e_{3}-e_{6}g_{-51}00000000000000000000000000000000000-g_{-63}0000-g_{-62}00000000000g_{-58}00g_{-56}000g_{-55}00g_{-54}00-g_{-51}g_{-51}0-g_{-51}g_{-51}0000g_{-48}00g_{-47}0000g_{-44}00g_{-42}000000g_{-37}000-g_{-34}00g_{-31}00-g_{-28}00g_{-25}00-g_{-21}0000-g_{-14}0-g_{-11}00-g_{-6}-g_{-3}00-2h_{8}-2h_{7}-2h_{6}-h_{5}-h_{4}-h_{3}00-g_{2}-g_{5}-g_{9}0-g_{12}000g_{24}g_{29}0
(0, -1, -1, -1, -1, -1, -2, -2)-e_{2}-e_{7}g_{-50}00000000000000000000000000000-g_{-64}000000000g_{-62}00000g_{-60}00000g_{-57}000000g_{-54}0000g_{-53}-g_{-50}g_{-50}000-g_{-50}g_{-50}00g_{-47}0000g_{-46}00g_{-43}0000g_{-41}0g_{-39}0000g_{-36}0g_{-34}000000000-g_{-24}0-g_{-22}00-g_{-17}0-g_{-15}0-g_{-10}00-g_{-7}-g_{-2}00-2h_{8}-2h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}00-g_{1}-g_{6}00-g_{13}00-g_{19}0-g_{25}0g_{35}
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(0, 1, 1, 1, 1, 2, 2, 2)e_{2}+e_{6}g_{54}-g_{-29}0g_{-18}0g_{-12}00g_{-5}g_{-1}02h_{8}+2h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}+h_{2}00g_{2}g_{6}0g_{10}0g_{14}0g_{17}00g_{21}0000g_{28}0-g_{30}000g_{34}0-g_{36}000000-g_{41}000-g_{44}00-g_{46}000-g_{48}000-g_{50}000-g_{51}0g_{54}-g_{54}00g_{54}-g_{54}00-g_{56}000-g_{57}000000-g_{60}00000-g_{62}0000000000g_{64}00000000000000000000000000000000000
(0, 0, 1, 1, 2, 2, 2, 2)e_{3}+e_{5}g_{55}0-g_{-23}-g_{-17}00g_{-9}g_{-4}g_{-2}02h_{8}+2h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}00g_{3}g_{5}000g_{13}00-g_{18}0g_{20}00-g_{25}0g_{27}000-g_{31}0g_{33}000-g_{37}0g_{39}0000-g_{42}0g_{44}0000-g_{47}0000000-g_{51}0-g_{52}000g_{55}-g_{55}g_{55}-g_{55}0000-g_{57}0-g_{58}0000-g_{59}0000000g_{62}00000g_{63}00000000000000000000000000000000000000000
(1, 1, 1, 1, 1, 2, 2, 2)e_{1}+e_{6}g_{56}g_{-24}g_{-18}0g_{-12}0g_{-5}002h_{8}+2h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{6}0g_{9}0g_{14}g_{16}00g_{21}g_{23}000g_{28}0000g_{34}-g_{35}00000-g_{40}0000-g_{44}0-g_{45}0000-g_{48}0-g_{49}0000-g_{51}00-g_{53}0000-g_{54}-g_{56}000g_{56}-g_{56}000000-g_{59}000000-g_{61}00000-g_{63}00000-g_{64}0000000000000000000000000000000000000000
(0, 1, 1, 1, 2, 2, 2, 2)e_{2}+e_{5}g_{57}-g_{-23}0g_{-11}0g_{-4}g_{-1}02h_{8}+2h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}0g_{2}g_{5}0g_{10}0g_{13}000g_{20}0-g_{24}00g_{27}0-g_{30}00g_{33}00-g_{36}00g_{39}00-g_{41}00g_{44}000-g_{46}000000-g_{50}00-g_{52}0000-g_{54}00-g_{55}0g_{57}-g_{57}0g_{57}-g_{57}000-g_{59}00-g_{60}000000-g_{62}00000000000g_{64}00000000000000000000000000000000000000000
(0, 0, 1, 2, 2, 2, 2, 2)e_{3}+e_{4}g_{58}0-g_{-16}-g_{-10}g_{-9}g_{-2}02h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}00g_{4}00-g_{11}g_{12}00-g_{18}g_{19}000-g_{25}g_{26}000-g_{31}g_{32}0000-g_{37}g_{38}0000-g_{42}g_{43}00000-g_{47}g_{48}00000-g_{51}g_{52}000000-g_{55}0000g_{58}0-g_{58}00000-g_{60}000000-g_{61}g_{62}00000g_{63}000000000000000000000000000000000000000000000000
(1, 1, 1, 1, 2, 2, 2, 2)e_{1}+e_{5}g_{59}g_{-17}g_{-11}0g_{-4}02h_{8}+2h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{5}g_{9}0g_{13}g_{16}00g_{20}000g_{27}-g_{29}000g_{33}-g_{35}000g_{39}0-g_{40}000g_{44}0-g_{45}000000-g_{49}000-g_{52}00-g_{53}000-g_{55}000-g_{56}000-g_{57}-g_{59}00g_{59}-g_{59}000000-g_{61}000000-g_{63}00000-g_{64}00000000000000000000000000000000000000000000000
(0, 1, 1, 2, 2, 2, 2, 2)e_{2}+e_{4}g_{60}-g_{-16}0g_{-3}g_{-1}2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}0g_{2}g_{4}00g_{12}0-g_{17}0g_{19}0-g_{24}0g_{26}00-g_{30}0g_{32}00-g_{36}0g_{38}000-g_{41}0g_{43}000-g_{46}0g_{48}0000-g_{50}0g_{52}0000-g_{54}0000000-g_{57}0-g_{58}0g_{60}-g_{60}g_{60}-g_{60}0000-g_{61}0-g_{62}000000000000g_{64}000000000000000000000000000000000000000000000000
(1, 1, 1, 2, 2, 2, 2, 2)e_{1}+e_{4}g_{61}g_{-10}g_{-3}02h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}g_{1}g_{4}g_{9}0g_{12}00g_{19}-g_{23}00g_{26}-g_{29}00g_{32}0-g_{35}00g_{38}0-g_{40}00g_{43}00-g_{45}00g_{48}00-g_{49}00g_{52}000-g_{53}000000-g_{56}00-g_{58}0000-g_{59}00-g_{60}-g_{61}0g_{61}-g_{61}000000-g_{63}000000-g_{64}000000000000000000000000000000000000000000000000000000
(0, 1, 2, 2, 2, 2, 2, 2)e_{2}+e_{3}g_{62}-g_{-9}g_{-1}2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}0g_{3}0-g_{10}g_{11}0-g_{17}g_{18}00-g_{24}g_{25}00-g_{30}g_{31}000-g_{36}g_{37}000-g_{41}g_{42}0000-g_{46}g_{47}0000-g_{50}g_{51}00000-g_{54}g_{55}00000-g_{57}g_{58}000000-g_{60}00g_{62}0-g_{62}00000-g_{63}0000000g_{64}0000000000000000000000000000000000000000000000000000000
(1, 1, 2, 2, 2, 2, 2, 2)e_{1}+e_{3}g_{63}g_{-2}2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}g_{1}g_{3}0g_{11}-g_{16}0g_{18}-g_{23}0g_{25}0-g_{29}0g_{31}0-g_{35}0g_{37}00-g_{40}0g_{42}00-g_{45}0g_{47}000-g_{49}0g_{51}000-g_{53}0g_{55}0000-g_{56}0g_{58}0000-g_{59}0000000-g_{61}0-g_{62}-g_{63}g_{63}-g_{63}000000-g_{64}00000000000000000000000000000000000000000000000000000000000000
(1, 2, 2, 2, 2, 2, 2, 2)e_{1}+e_{2}g_{64}2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}g_{2}-g_{9}g_{10}-g_{16}g_{17}0-g_{23}g_{24}0-g_{29}g_{30}00-g_{35}g_{36}00-g_{40}g_{41}000-g_{45}g_{46}000-g_{49}g_{50}0000-g_{53}g_{54}0000-g_{56}g_{57}00000-g_{59}g_{60}00000-g_{61}g_{62}000000-g_{63}00-g_{64}0000000000000000000000000000000000000000000000000000000000000000000000
We define the symmetric Cartan matrix
by requesting that the entry in the i-th row and j-th column
be the scalar product of the i^th and j^th roots. The symmetric Cartan matrix is:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1\\ \end{pmatrix}\)
Let the (i, j)^{th} entry of the symmetric Cartan matrix be a_{ij}.
Then we define the co-symmetric Cartan matrix as the matrix whose (i, j)^{th} entry equals 4*a_{ij}/(a_{ii}*a_{jj}). In other words, the co-symmetric Cartan matrix is the symmetric Cartan matrix of the dual root system. The co-symmetric Cartan matrix equals:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -2\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 4\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1
Half sum of positive roots: (15/2, 14, 39/2, 24, 55/2, 30, 63/2, 32)= \(\displaystyle 15/2\varepsilon_{1}+13/2\varepsilon_{2}+11/2\varepsilon_{3}+9/2\varepsilon_{4}+7/2\varepsilon_{5}+5/2\varepsilon_{6}+3/2\varepsilon_{7}+1/2\varepsilon_{8}\)
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):
(1, 1, 1, 1, 1, 1, 1, 1) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 2, 2, 2, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3, 3, 3, 3) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 4, 4, 4, 4) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5, 5, 5, 5) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
(1, 2, 3, 4, 5, 6, 6, 6) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}\)
(1, 2, 3, 4, 5, 6, 7, 7) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}\)
(1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}+1/2\varepsilon_{8}\)
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, 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) = \(\displaystyle \varepsilon_{8}\)