On Chevalley–Shephard–Todd’s theorem in positive characteristic

Abstract

Let G be a finite group acting linearly on the vector space V over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants and the direct summand property holds if there is a surjective k[V]G-linear map π: k[V]→k[V]G. The following Chevalley–Shephard–Todd type theorem is proved. Suppose V is an irreducible kG-representation, then the action is coregular if and only if G is generated by pseudo-reflections and the direct summand property holds.