# BettiCharacters -- finite group characters on free resolutions and graded modules

## Description

This package contains functions for computing characters of free resolutions and graded modules equipped with the action of a finite group.

Let $R$ be a positively graded polynomial ring over a field $\Bbbk$, and $M$ a finitely generated graded $R$-module. Suppose $G$ is a finite group whose order is not divisible by the characteristic of $\Bbbk$. Assume $G$ acts $\Bbbk$-linearly on $R$ and $M$ by preserving degrees, and distributing over $R$-multiplication. If $F_\bullet$ is a minimal free resolution of $M$, and $\mathfrak{m}$ denotes the maximal ideal generated by the variables of $R$, then each $F_i / \mathfrak{m}F_i$ is a graded $G$-representation. We call the characters of the representations $F_i / \mathfrak{m}F_i$ the Betti characters of $M$, since evaluating them at the identity element of $G$ returns the usual Betti numbers of $M$. Moreover, the graded components of $M$ are also $G$-representations.

This package provides functions to compute the Betti characters and the characters of graded components of $M$ based on the algorithms in F. Galetto - Finite group characters on free resolutions. The package is designed to be independent of the group; the user provides matrices for the group actions and character tables (to decompose characters into irreducibles). See the menu below for using this package to compute some examples from the literature.

#### Version history:

• 1.0: Initial version. Includes computation of actions and Betti characters.
• 2.0: Introduces character tables, decompositions, and other methods for characters.

## Version

This documentation describes version 2.0 of BettiCharacters.

## Source code

The source code from which this documentation is derived is in the file BettiCharacters.m2.

## For the programmer

The object BettiCharacters is .

#### Defining and computing actions

• action -- define finite group action
• actors -- group elements of an action