The Direct Summand Property in Modular Invariant Theory

Abstract

In nonmodular invariant theory of finite groups, the invariant ring is always a direct summand of the full polynomial ring. This is no longer generally true in modular invariant theory, but nevertheless interesting examples are known where this happens. We give useful characterisations of the direct summand property in terms of the image of a twisted transfer map. For example, for p-groups acting in characteristic p the direct summand property holds if and only if the image of the ordinary transfer is a principal ideal; and in that case the group is generated by transvections. We also extend some known results in the nonmodular case to where only the direct summand property is assumed, e.g., the invariant ring is always generated by its elements of degree at most the order of the group.