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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 \ell being the rank of the reflection group, the degrees d_1 \leq d_2 \leq \ldots \leq d_\ell 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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