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

TL;DR

This paper studies the structure of invariant rings for a specific class of $p$-groups in characteristic $p$, proving they are complete intersections and confirming conjectures about their generators and polynomial nature.

Contribution

It establishes that for generalised Nakajima groups, the Hilbert ideal is a complete intersection and confirms a conjecture relating direct summands to polynomial invariants.

Findings

Hilbert ideal is a complete intersection for these groups

Invariant ring $S^G$ is polynomial if it is a direct summand

Generators of the Hilbert ideal have degree at most |G|

Abstract

Let $p$ be a prime number, $\Bbbk$ a field of characteristic $p$ and $G$ a finite $p$-group. Let $V$ be a finite-dimensional linear representation of $G$ over $\Bbbk$. Write $S = \mathrm{Sym} V^*$. 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}