On Hilbert ideals for a class of $p$-groups in characteristic $p$

Abstract

Let $p$ be a prime number, $\mathbb{k}$ a field of characteristic $p$ and $G$ a finite $p$-group. Let $V$ be a finite-dimensional linear representation of $G$ over $\mathbb{k}$. Write $S=\mathrm{Sym}{V}^{\ast }$. For a class of $p$-groups which we call generalised Nakajima groups, we prove the following: \begin{enumerate} \item The Hilbert ideal is a complete intersection. As a consequence, for the case of generalised Nakajima groups, we prove a conjecture of Shank and Wehlau (reformulated by Broer) that asserts that if the invariant subring $S^G$ is a direct summand of $S$ as $S^G$-modules then $S^G$ is a polynomial ring. \item The Hilbert ideal has a generating set with elements of degree at most $|G|$. This bound is conjectured by Derksen and Kemper. \end{enumerate}