An algorithm for invariants of elementary abelian groups

TL;DR

This paper introduces a new, more efficient algorithm for computing invariant rings of elementary abelian groups acting on polynomial rings over characteristic zero fields.

Contribution

The authors develop a two-step algorithm that generates seed invariants and expands them using lattice structures, outperforming existing methods like Macaulay2.

Findings

Algorithm computes invariants more quickly than existing methods.

Uses kernel of weight matrix to generate seed invariants.

Exploits lattice structure of invariants modulo p.

Abstract

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many minimal generators, but efficiently finding a generating set is not a trivial task. We present a new algorithm for computing the invariant ring for elementary abelian groups acting on polynomial rings with complex coefficients (or any other field of characteristic zero). We follow a two-step process: first we generate a collection of $n-k$ "seed" invariants by calculating the kernel of a weight matrix that encodes our action. After we find the seeds, we "grow" them into a generating set for the invariant ring by exploiting the lattice structure of invariants modulo $p$. Our algorithm performs better than the one currently available in Macaulay2, allowing us to compute invariants more quickly in this setting.