Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - June 2012 - Volume 55, Issue 02  

Research Article

Invariants of hyperplane groups and vanishing ideals of finite sets of points

H. E. A. Campbella1 and Jianjun Chuaia1

a1 Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada (eddy@unb.ca; jchuai@unb.ca)

Abstract

We define a hyperplane group to be a finite group generated by reflections fixing a single hyperplane pointwise. Landweber and Stong proved that the invariant ring of a hyperplane group is again a polynomial ring in any characteristic. Recently, Hartmann and Shepler gave a constructive proof of this result. By their algorithm, one can always construct generators that are additive. In this paper, we study hyperplane groups of order a power of a prime p in characteristic p and give a slightly different construction of the generators than Hartmann and Shepler. We then show that such generators have a particular form. Furthermore, we show that if the group is defined by a finite additive subgroup W ⊆ $W\subseteq\mathbb{F}^n$, the vanishing ideal of W is generated by polynomials obtained from a set of generators of the invariant ring that are additive. Finally, we give a shorter proof of the fact that the module of the invariant differential 1-forms is free in our situation.

(Received October 31 2008)

Keywords

  • hyperplane group;
  • invariant ring;
  • vanishing ideal;
  • invariant differential 1-form

2010 Mathematics subject classification

  • Primary 13A50;
  • 14R99