Complex Reflection Group - Degrees

Degrees

Shephard and Todd proved that a finite group acting on a complex vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring (Chevalley–Shephard–Todd theorem). For being the rank of the reflection group, the degrees of the generators of the ring of invariants are called degrees of W and are listed in the column above headed "degrees". They also showed that many other invariants of the group are determined by the degrees as follows:

  • The center of an irreducible reflection group is cyclic of order equal to the greatest common divisor of the degrees.
  • The order of a complex reflection group is the product of its degrees.
  • The number of reflections is the sum of the degrees minus the rank.
  • An irreducible complex reflection group comes from a real reflection group if and only if it has an invariant of degree 2.
  • The degrees di satisfy the formula

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