sp(14), type \(C^{1}_7\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: C^{1}_7.
Weyl group size: 645120.
A drawing of the root system in its corresponding Coxeter plane. Computations were carried out as explained by John Stembridge.
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The grey lines are the edges of the Weyl chamber.
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The root system has 98 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-2, -2, -2, -2, -2, -2, -1)-2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(-1, -2, -2, -2, -2, -2, -1)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, -2, -2, -2, -2, -2, -1)-2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -2, -2, -1)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -2, -2, -2, -2, -1)-e_{2}-e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(-1, -1, -1, -2, -2, -2, -1)-e_{1}-e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -2, -2, -2, -2, -1)-2e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -2, -2, -2, -1)-e_{2}-e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}\)
(-1, -1, -1, -1, -2, -2, -1)-e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}\)
(0, 0, -1, -2, -2, -2, -1)-e_{3}-e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(0, -1, -1, -1, -2, -2, -1)-e_{2}-e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}\)
(-1, -1, -1, -1, -1, -2, -1)-e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}\)
(0, 0, 0, -2, -2, -2, -1)-2e_{4}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -2, -2, -1)-e_{3}-e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(0, -1, -1, -1, -1, -2, -1)-e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}\)
(-1, -1, -1, -1, -1, -1, -1)-e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}\)
(0, 0, 0, -1, -2, -2, -1)-e_{4}-e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(0, 0, -1, -1, -1, -2, -1)-e_{3}-e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(0, -1, -1, -1, -1, -1, -1)-e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}\)
(-1, -1, -1, -1, -1, -1, 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, -2, -2, -1)-2e_{5}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -2, -1)-e_{4}-e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(0, 0, -1, -1, -1, -1, -1)-e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(0, -1, -1, -1, -1, -1, 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)-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, -1, -2, -1)-e_{5}-e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{5}s_{7}s_{6}\)
(0, 0, 0, -1, -1, -1, -1)-e_{4}-e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}\)
(0, 0, -1, -1, -1, -1, 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)-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)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -2, -1)-2e_{6}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, -1)-e_{5}-e_{7}\(s_{5}s_{7}s_{6}s_{5}s_{7}\)
(0, 0, 0, -1, -1, -1, 0)-e_{4}+e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, -1)-e_{6}-e_{7}\(s_{7}s_{6}s_{7}\)
(0, 0, 0, 0, -1, -1, 0)-e_{5}+e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, -1)-2e_{7}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0)-e_{6}+e_{7}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0)e_{6}-e_{7}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1)2e_{7}\(s_{7}\)
(1, 1, 0, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0)e_{5}-e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1)e_{6}+e_{7}\(s_{7}s_{6}s_{7}\)
(1, 1, 1, 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)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0)e_{4}-e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1)e_{5}+e_{7}\(s_{5}s_{7}s_{6}s_{5}s_{7}\)
(0, 0, 0, 0, 0, 2, 1)2e_{6}\(s_{6}s_{7}s_{6}\)
(1, 1, 1, 1, 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)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)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)e_{4}+e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}\)
(0, 0, 0, 0, 1, 2, 1)e_{5}+e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{5}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 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)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)e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(0, 0, 0, 1, 1, 2, 1)e_{4}+e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(0, 0, 0, 0, 2, 2, 1)2e_{5}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 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)e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}\)
(0, 0, 1, 1, 1, 2, 1)e_{3}+e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(0, 0, 0, 1, 2, 2, 1)e_{4}+e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 1)e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}\)
(0, 1, 1, 1, 1, 2, 1)e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}\)
(0, 0, 1, 1, 2, 2, 1)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(0, 0, 0, 2, 2, 2, 1)2e_{4}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 1, 2, 1)e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}\)
(0, 1, 1, 1, 2, 2, 1)e_{2}+e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}\)
(0, 0, 1, 2, 2, 2, 1)e_{3}+e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 2, 2, 1)e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}\)
(0, 1, 1, 2, 2, 2, 1)e_{2}+e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 2, 2, 2, 2, 1)2e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 2, 1)e_{1}+e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(0, 1, 2, 2, 2, 2, 1)e_{2}+e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 2, 2, 1)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 2, 2, 2, 2, 2, 1)2e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(1, 2, 2, 2, 2, 2, 1)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(2, 2, 2, 2, 2, 2, 1)2e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-2, -2, -2, -2, -2, -2, -1), (-1, -2, -2, -2, -2, -2, -1), (0, -2, -2, -2, -2, -2, -1), (-1, -1, -2, -2, -2, -2, -1), (0, -1, -2, -2, -2, -2, -1), (-1, -1, -1, -2, -2, -2, -1), (0, 0, -2, -2, -2, -2, -1), (0, -1, -1, -2, -2, -2, -1), (-1, -1, -1, -1, -2, -2, -1), (0, 0, -1, -2, -2, -2, -1), (0, -1, -1, -1, -2, -2, -1), (-1, -1, -1, -1, -1, -2, -1), (0, 0, 0, -2, -2, -2, -1), (0, 0, -1, -1, -2, -2, -1), (0, -1, -1, -1, -1, -2, -1), (-1, -1, -1, -1, -1, -1, -1), (0, 0, 0, -1, -2, -2, -1), (0, 0, -1, -1, -1, -2, -1), (0, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, 0), (0, 0, 0, 0, -2, -2, -1), (0, 0, 0, -1, -1, -2, -1), (0, 0, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -1, -1, 0, 0), (0, 0, 0, 0, -1, -2, -1), (0, 0, 0, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, 0, -2, -1), (0, 0, 0, 0, -1, -1, -1), (0, 0, 0, -1, -1, -1, 0), (0, 0, -1, -1, -1, 0, 0), (0, -1, -1, -1, 0, 0, 0), (-1, -1, -1, 0, 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, -1, 0, 0, 0, 0), (-1, -1, 0, 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, 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, 0, 0), (0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 1, 1), (1, 1, 1, 0, 0, 0, 0), (0, 1, 1, 1, 0, 0, 0), (0, 0, 1, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 0, 1, 1, 1), (0, 0, 0, 0, 0, 2, 1), (1, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 2, 1), (1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1, 1), (0, 0, 0, 1, 1, 2, 1), (0, 0, 0, 0, 2, 2, 1), (1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1, 1), (0, 0, 1, 1, 1, 2, 1), (0, 0, 0, 1, 2, 2, 1), (1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 1, 2, 1), (0, 0, 1, 1, 2, 2, 1), (0, 0, 0, 2, 2, 2, 1), (1, 1, 1, 1, 1, 2, 1), (0, 1, 1, 1, 2, 2, 1), (0, 0, 1, 2, 2, 2, 1), (1, 1, 1, 1, 2, 2, 1), (0, 1, 1, 2, 2, 2, 1), (0, 0, 2, 2, 2, 2, 1), (1, 1, 1, 2, 2, 2, 1), (0, 1, 2, 2, 2, 2, 1), (1, 1, 2, 2, 2, 2, 1), (0, 2, 2, 2, 2, 2, 1), (1, 2, 2, 2, 2, 2, 1), (2, 2, 2, 2, 2, 2, 1) The resulting Lie bracket pairing table follows.
Type C^{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.
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(0, 1, 1, 1, 1, 0, 0)e_{2}-e_{6}g_{21}0-g_{-38}-g_{-35}0-g_{-32}00-g_{-28}00-g_{-24}000-2g_{-19}000-g_{-13}0000-g_{-6}g_{-1}0002h_{5}+2h_{4}+2h_{3}+2h_{2}0000g_{2}-g_{5}000g_{9}0-g_{11}000g_{15}00-g_{16}01/2g_{21}-1/2g_{21}00-1/2g_{21}1/2g_{21}0-g_{25}0000g_{26}000000g_{31}00000g_{35}0000g_{39}000g_{42}000g_{45}002g_{47}00g_{48}00000000000
(0, 0, 1, 1, 1, 1, 0)e_{3}-e_{7}g_{22}000-g_{-34}-g_{-31}0-g_{-27}00-g_{-23}000-g_{-18}000-g_{-13}0g_{-8}00-2g_{-7}g_{-2}0002h_{6}+2h_{5}+2h_{4}+2h_{3}0000g_{3}-g_{6}000g_{10}0-g_{12}000g_{16}00-g_{17}0001/2g_{22}-1/2g_{22}00-1/2g_{22}g_{22}0-g_{26}0000g_{27}-g_{30}0000g_{32}0000g_{36}0000g_{40}0002g_{43}000g_{45}00g_{46}000000000000000
(0, 0, 0, 1, 1, 1, 1)e_{4}+e_{7}g_{23}00000g_{-30}0g_{-26}0g_{-22}00g_{-17}00g_{-14}g_{-12}0g_{-9}00g_{-6}g_{-3}0002h_{7}+2h_{6}+2h_{5}+2h_{4}0000g_{4}-2g_{7}000g_{11}0-g_{13}000g_{17}00-g_{18}000001/2g_{23}-1/2g_{23}01/2g_{23}-g_{23}00-g_{27}00-g_{28}00-g_{31}00-g_{33}0-g_{34}00-2g_{37}0000-g_{40}000-g_{42}000-g_{44}0000000000000000000
(0, 0, 0, 0, 1, 2, 1)e_{5}+e_{6}g_{24}00000000g_{-25}0g_{-21}g_{-20}0g_{-16}g_{-15}0g_{-11}g_{-10}00g_{-5}g_{-4}0002h_{7}+4h_{6}+2h_{5}0000g_{5}g_{6}0000g_{12}-g_{13}00000-g_{18}-2g_{19}00000001/2g_{24}0-1/2g_{24}0000-g_{28}-2g_{29}0000-g_{32}-g_{33}000-g_{35}-g_{36}000-g_{38}-g_{39}000-g_{41}000000000000000000000000
(1, 1, 1, 1, 1, 0, 0)e_{1}-e_{6}g_{25}-g_{-38}-g_{-35}0-g_{-32}0-g_{-28}00-g_{-24}00-2g_{-19}000-g_{-13}000-g_{-6}00002h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}000g_{1}-g_{5}000g_{8}0-g_{11}00g_{14}00-g_{16}00g_{20}000-g_{21}-1/2g_{25}000-1/2g_{25}1/2g_{25}000000g_{30}000000g_{34}00000g_{38}0000g_{41}000g_{44}000g_{46}00g_{48}002g_{49}00000000000
(0, 1, 1, 1, 1, 1, 0)e_{2}-e_{7}g_{26}0-g_{-34}-g_{-31}0-g_{-27}00-g_{-23}00-g_{-18}000-g_{-13}000-2g_{-7}g_{-1}0002h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}000g_{2}-g_{6}000g_{9}0-g_{12}00g_{15}00-g_{17}00g_{21}000-g_{22}01/2g_{26}-1/2g_{26}000-1/2g_{26}g_{26}-g_{30}00000g_{31}00000g_{35}0000g_{39}0000g_{42}000g_{45}0002g_{47}00g_{48}000000000000000
(0, 0, 1, 1, 1, 1, 1)e_{3}+e_{7}g_{27}000g_{-30}g_{-26}0g_{-22}00g_{-17}000g_{-12}0g_{-8}0g_{-6}g_{-2}0002h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}000g_{3}-2g_{7}000g_{10}0-g_{13}00g_{16}00-g_{18}00g_{22}000-g_{23}0001/2g_{27}-1/2g_{27}001/2g_{27}-g_{27}0-g_{31}000-g_{32}0-g_{34}000-g_{36}0000-g_{40}0000-2g_{43}000-g_{45}000-g_{46}0000000000000000000
(0, 0, 0, 1, 1, 2, 1)e_{4}+e_{6}g_{28}00000g_{-25}0g_{-21}0g_{-16}0g_{-14}g_{-11}0g_{-9}0g_{-5}g_{-3}0002h_{7}+4h_{6}+2h_{5}+2h_{4}000g_{4}g_{6}000g_{11}0-g_{13}000g_{17}0-2g_{19}0000-g_{23}0-g_{24}000001/2g_{28}-1/2g_{28}1/2g_{28}-1/2g_{28}000-g_{32}0-g_{33}000-g_{35}0-2g_{37}00-g_{38}0-g_{40}0000-g_{42}000-g_{44}000000000000000000000000
(0, 0, 0, 0, 2, 2, 1)2e_{5}g_{29}00000000g_{-20}0g_{-15}00g_{-10}00g_{-4}000h_{7}+2h_{6}+2h_{5}0000g_{5}00000g_{12}00000-g_{18}000000-g_{24}0000000g_{29}-g_{29}00000-g_{33}00000-g_{36}0000-g_{39}0000-g_{41}00000000000000000000000000000
(1, 1, 1, 1, 1, 1, 0)e_{1}-e_{7}g_{30}-g_{-34}-g_{-31}0-g_{-27}0-g_{-23}00-g_{-18}00-g_{-13}000-2g_{-7}0002h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}000g_{1}-g_{6}00g_{8}0-g_{12}00g_{14}00-g_{17}0g_{20}000-g_{22}0g_{25}0000-g_{26}-1/2g_{30}0000-1/2g_{30}g_{30}000000g_{34}00000g_{38}0000g_{41}0000g_{44}000g_{46}000g_{48}002g_{49}000000000000000
(0, 1, 1, 1, 1, 1, 1)e_{2}+e_{7}g_{31}0g_{-30}g_{-26}0g_{-22}00g_{-17}00g_{-12}000g_{-6}g_{-1}002h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}000g_{2}-2g_{7}00g_{9}0-g_{13}00g_{15}00-g_{18}0g_{21}000-g_{23}0g_{26}0000-g_{27}01/2g_{31}-1/2g_{31}0001/2g_{31}-g_{31}-g_{34}0000-g_{35}00000-g_{39}0000-g_{42}0000-g_{45}000-2g_{47}000-g_{48}0000000000000000000
(0, 0, 1, 1, 1, 2, 1)e_{3}+e_{6}g_{32}000g_{-25}g_{-21}0g_{-16}00g_{-11}0g_{-8}0g_{-5}g_{-2}002h_{7}+4h_{6}+2h_{5}+2h_{4}+2h_{3}000g_{3}g_{6}00g_{10}0-g_{13}00g_{16}00-2g_{19}00g_{22}00-g_{24}000-g_{27}00-g_{28}0001/2g_{32}-1/2g_{32}01/2g_{32}-1/2g_{32}00-g_{35}00-g_{36}00-g_{38}00-g_{40}0000-2g_{43}0000-g_{45}000-g_{46}000000000000000000000000
(0, 0, 0, 1, 2, 2, 1)e_{4}+e_{5}g_{33}00000g_{-20}0g_{-15}g_{-14}g_{-10}g_{-9}0g_{-4}g_{-3}002h_{7}+4h_{6}+4h_{5}+2h_{4}000g_{4}g_{5}000g_{11}g_{12}0000g_{17}-g_{18}0000-g_{23}-g_{24}00000-g_{28}-2g_{29}000001/2g_{33}0-1/2g_{33}0000-g_{36}-2g_{37}0000-g_{39}-g_{40}000-g_{41}-g_{42}0000-g_{44}00000000000000000000000000000
(1, 1, 1, 1, 1, 1, 1)e_{1}+e_{7}g_{34}g_{-30}g_{-26}0g_{-22}0g_{-17}00g_{-12}00g_{-6}0002h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}00g_{1}-2g_{7}00g_{8}0-g_{13}0g_{14}00-g_{18}0g_{20}000-g_{23}g_{25}0000-g_{27}g_{30}00000-g_{31}-1/2g_{34}00001/2g_{34}-g_{34}00000-g_{38}00000-g_{41}0000-g_{44}0000-g_{46}000-g_{48}000-2g_{49}0000000000000000000
(0, 1, 1, 1, 1, 2, 1)e_{2}+e_{6}g_{35}0g_{-25}g_{-21}0g_{-16}00g_{-11}00g_{-5}g_{-1}002h_{7}+4h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}00g_{2}g_{6}00g_{9}0-g_{13}0g_{15}00-2g_{19}0g_{21}000-g_{24}0g_{26}000-g_{28}00-g_{31}000-g_{32}01/2g_{35}-1/2g_{35}001/2g_{35}-1/2g_{35}0-g_{38}000-g_{39}00000-g_{42}0000-g_{45}0000-2g_{47}000-g_{48}000000000000000000000000
(0, 0, 1, 1, 2, 2, 1)e_{3}+e_{5}g_{36}000g_{-20}g_{-15}0g_{-10}0g_{-8}g_{-4}g_{-2}002h_{7}+4h_{6}+4h_{5}+2h_{4}+2h_{3}00g_{3}g_{5}00g_{10}0g_{12}00g_{16}0-g_{18}000g_{22}0-g_{24}000-g_{27}0-2g_{29}0000-g_{32}0-g_{33}0001/2g_{36}-1/2g_{36}1/2g_{36}-1/2g_{36}000-g_{39}0-g_{40}000-g_{41}0-2g_{43}0000-g_{45}0000-g_{46}00000000000000000000000000000
(0, 0, 0, 2, 2, 2, 1)2e_{4}g_{37}00000g_{-14}0g_{-9}0g_{-3}00h_{7}+2h_{6}+2h_{5}+2h_{4}000g_{4}0000g_{11}0000g_{17}00000-g_{23}00000-g_{28}000000-g_{33}00000g_{37}-g_{37}00000-g_{40}00000-g_{42}0000-g_{44}00000000000000000000000000000000000
(1, 1, 1, 1, 1, 2, 1)e_{1}+e_{6}g_{38}g_{-25}g_{-21}0g_{-16}0g_{-11}00g_{-5}002h_{7}+4h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}00g_{1}g_{6}0g_{8}0-g_{13}0g_{14}00-2g_{19}g_{20}000-g_{24}g_{25}0000-g_{28}g_{30}0000-g_{32}0-g_{34}0000-g_{35}-1/2g_{38}0001/2g_{38}-1/2g_{38}00000-g_{41}00000-g_{44}0000-g_{46}0000-g_{48}000-2g_{49}000000000000000000000000
(0, 1, 1, 1, 2, 2, 1)e_{2}+e_{5}g_{39}0g_{-20}g_{-15}0g_{-10}00g_{-4}g_{-1}02h_{7}+4h_{6}+4h_{5}+2h_{4}+2h_{3}+2h_{2}00g_{2}g_{5}0g_{9}0g_{12}0g_{15}00-g_{18}0g_{21}00-g_{24}00g_{26}00-2g_{29}00-g_{31}00-g_{33}000-g_{35}00-g_{36}01/2g_{39}-1/2g_{39}01/2g_{39}-1/2g_{39}00-g_{41}00-g_{42}00000-g_{45}0000-2g_{47}0000-g_{48}00000000000000000000000000000
(0, 0, 1, 2, 2, 2, 1)e_{3}+e_{4}g_{40}000g_{-14}g_{-9}g_{-8}g_{-3}g_{-2}02h_{7}+4h_{6}+4h_{5}+4h_{4}+2h_{3}00g_{3}g_{4}00g_{10}g_{11}000g_{16}g_{17}000g_{22}-g_{23}0000-g_{27}-g_{28}0000-g_{32}-g_{33}00000-g_{36}-2g_{37}0001/2g_{40}0-1/2g_{40}0000-g_{42}-2g_{43}0000-g_{44}-g_{45}0000-g_{46}00000000000000000000000000000000000
(1, 1, 1, 1, 2, 2, 1)e_{1}+e_{5}g_{41}g_{-20}g_{-15}0g_{-10}0g_{-4}002h_{7}+4h_{6}+4h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}g_{5}0g_{8}0g_{12}g_{14}00-g_{18}g_{20}000-g_{24}g_{25}000-2g_{29}0g_{30}000-g_{33}0-g_{34}000-g_{36}00-g_{38}000-g_{39}-1/2g_{41}001/2g_{41}-1/2g_{41}00000-g_{44}00000-g_{46}0000-g_{48}0000-2g_{49}00000000000000000000000000000
(0, 1, 1, 2, 2, 2, 1)e_{2}+e_{4}g_{42}0g_{-14}g_{-9}0g_{-3}g_{-1}02h_{7}+4h_{6}+4h_{5}+4h_{4}+2h_{3}+2h_{2}0g_{2}g_{4}0g_{9}0g_{11}0g_{15}0g_{17}00g_{21}0-g_{23}00g_{26}0-g_{28}000-g_{31}0-g_{33}000-g_{35}0-2g_{37}0000-g_{39}0-g_{40}01/2g_{42}-1/2g_{42}1/2g_{42}-1/2g_{42}000-g_{44}0-g_{45}00000-2g_{47}0000-g_{48}00000000000000000000000000000000000
(0, 0, 2, 2, 2, 2, 1)2e_{3}g_{43}000g_{-8}g_{-2}0h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}00g_{3}000g_{10}000g_{16}0000g_{22}0000-g_{27}00000-g_{32}00000-g_{36}000000-g_{40}000g_{43}-g_{43}00000-g_{45}00000-g_{46}00000000000000000000000000000000000000000
(1, 1, 1, 2, 2, 2, 1)e_{1}+e_{4}g_{44}g_{-14}g_{-9}0g_{-3}02h_{7}+4h_{6}+4h_{5}+4h_{4}+2h_{3}+2h_{2}+2h_{1}0g_{1}g_{4}g_{8}0g_{11}g_{14}00g_{17}g_{20}00-g_{23}0g_{25}00-g_{28}0g_{30}00-g_{33}00-g_{34}00-2g_{37}00-g_{38}00-g_{40}000-g_{41}00-g_{42}-1/2g_{44}01/2g_{44}-1/2g_{44}00000-g_{46}00000-g_{48}0000-2g_{49}00000000000000000000000000000000000
(0, 1, 2, 2, 2, 2, 1)e_{2}+e_{3}g_{45}0g_{-8}g_{-2}g_{-1}2h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+2h_{2}0g_{2}g_{3}0g_{9}g_{10}00g_{15}g_{16}00g_{21}g_{22}000g_{26}-g_{27}000-g_{31}-g_{32}0000-g_{35}-g_{36}0000-g_{39}-g_{40}00000-g_{42}-2g_{43}01/2g_{45}0-1/2g_{45}0000-g_{46}-2g_{47}00000-g_{48}00000000000000000000000000000000000000000
(1, 1, 2, 2, 2, 2, 1)e_{1}+e_{3}g_{46}g_{-8}g_{-2}02h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+2h_{2}+2h_{1}g_{1}g_{3}g_{8}0g_{10}g_{14}0g_{16}0g_{20}0g_{22}0g_{25}0-g_{27}00g_{30}0-g_{32}00-g_{34}0-g_{36}000-g_{38}0-g_{40}000-g_{41}0-2g_{43}0000-g_{44}0-g_{45}-1/2g_{46}1/2g_{46}-1/2g_{46}00000-g_{48}00000-2g_{49}00000000000000000000000000000000000000000
(0, 2, 2, 2, 2, 2, 1)2e_{2}g_{47}0g_{-1}h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}0g_{2}00g_{9}00g_{15}000g_{21}000g_{26}0000-g_{31}0000-g_{35}00000-g_{39}00000-g_{42}000000-g_{45}0g_{47}-g_{47}00000-g_{48}000000000000000000000000000000000000000000000000
(1, 2, 2, 2, 2, 2, 1)e_{1}+e_{2}g_{48}g_{-1}2h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}g_{1}g_{2}g_{8}g_{9}0g_{14}g_{15}0g_{20}g_{21}00g_{25}g_{26}00g_{30}-g_{31}000-g_{34}-g_{35}000-g_{38}-g_{39}0000-g_{41}-g_{42}0000-g_{44}-g_{45}00000-g_{46}-2g_{47}0-1/2g_{48}00000-2g_{49}000000000000000000000000000000000000000000000000
(2, 2, 2, 2, 2, 2, 1)2e_{1}g_{49}h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}g_{1}0g_{8}0g_{14}00g_{20}00g_{25}000g_{30}000-g_{34}0000-g_{38}0000-g_{41}00000-g_{44}00000-g_{46}000000-g_{48}-g_{49}0000000000000000000000000000000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0\\ 0 & -1/2 & 1 & -1/2 & 0 & 0 & 0\\ 0 & 0 & -1/2 & 1 & -1/2 & 0 & 0\\ 0 & 0 & 0 & -1/2 & 1 & -1/2 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0 & 0 & 0\\ 0 & -2 & 4 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & 4 & -2 & 0 & 0\\ 0 & 0 & 0 & -2 & 4 & -2 & 0\\ 0 & 0 & 0 & 0 & -2 & 4 & -2\\ 0 & 0 & 0 & 0 & 0 & -2 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1/32
Half sum of positive roots: (7, 13, 18, 22, 25, 27, 14)= \(\displaystyle 7\varepsilon_{1}+6\varepsilon_{2}+5\varepsilon_{3}+4\varepsilon_{4}+3\varepsilon_{5}+2\varepsilon_{6}+\varepsilon_{7}\)
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, 1/2) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 2, 2, 1) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3, 3, 3/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 4, 4, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5, 5, 5/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
(1, 2, 3, 4, 5, 6, 3) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}\)
(1, 2, 3, 4, 5, 6, 7/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}\)
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) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)
(0, 0, 0, 0, 0, 0, 1) = \(\displaystyle 2\varepsilon_{7}\)