Setting the scene for Betti characters

TL;DR

This paper introduces a Macaulay2 package that computes characters of finite groups acting on free resolutions and modules over polynomial rings, aiding the understanding of these actions in algebraic contexts.

Contribution

The paper presents a new computational tool for analyzing group actions on algebraic structures, specifically through characters of modules and resolutions, which was previously difficult to automate.

Findings

Provides a Macaulay2 package for character computation

Enables analysis of group actions on free resolutions

Facilitates understanding of module structures in algebra

Abstract

Finite group actions on free resolutions and modules arise naturally in many interesting examples. Understanding these actions amounts to describing the terms of a free resolution or the graded components of a module as group representations which, in the non modular case, are completely determined by their characters. With this goal in mind, we introduce a Macaulay2 package for computing characters of finite groups on free resolutions and graded components of finitely generated graded modules over polynomial rings.