# 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 may provide the necessary information about the group action in the form of matrices and/or substitutions into the variables of the ring. See the menu below for using this package to compute some examples from the literature.

## Version

This documentation describes version 1.0 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 actions

• action -- define finite group action

#### Computing actions and characters

• actors -- group elements of an action
• character -- compute characters of finite group action

#### Examples from the literature

• Example 1 -- Specht ideals / subspace arrangements
• Example 2 -- Symbolic powers of star configurations
• Example 3 -- Klein configuration of points