Invariants of hyperplane groups and vanishing ideals of finite sets of points (Q2891989)

A \textit{hyperplane group} \(G\) is a finite linear group acting on an \(n\)-dimensional vector space \(V\) over a field \(\mathbb{F}\) such that every group element is fixing the same hyperplane pointwise. This paper describes the ring of polynomial invariants \(\mathbb{F}[V]^G\) in the modular case, i.e. when the characteristic of \(\mathbb{F}\) divides the order of \(G\). Earlier it was proved by \textit{P. S. Landweber} and \textit{R. E. Stong} that in this case \(\mathbb{F}[V]^G\) must be a polynomial ring [Lect. Notes Math. 1240, 259--274 (1987; Zbl 0623.55007)] and subsequently \textit{J. Hartmann} and \textit{A. V. Shepler} gave an algorithm for constructing its generators [Trans. Am. Math. Soc. 360, No. 1, 123--133 (2008; Zbl 1129.13003)]. Based on a refined version of this algorithm, the explicit form of these generators is given in Theorem 2.3 in this paper.NEWLINENEWLINEAs a first application it is shown that by identifying \(G\) with a finite additive subgroup \(W \subset \mathbb{F}^n\), the vanishing ideal of \(W\) will be generated by polynomials which are obtained from the generators of \(\mathbb{F}[V]^G\) by a simple substitution. As a second application an explicit list of basis elements of \((\Omega^1)^G\), the \(\mathbb{F}[V]^G\)-module of the \(G\)-invariant differential \(1\)-forms is given in Theorem 2.6., which leads to a simpler proof of the fact that this module is free, which was originally proved by \textit{J. Hartmann} and \textit{A. V. Shepler}, Theorem 7 in [Math. Res. Lett. 14, No. 5--6, 955--971 (2007; Zbl 1151.58300)].