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


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.
  • 2.1: Adds equality checks for actions and characters. Contains several small improvements to the code and documentation, including a new multigraded example.


Certification a gold star

Version 2.1 of this package was accepted for publication in volume 13 of Journal of Software for Algebra and Geometry on 2023-05-30, in the article Setting the scene for Betti characters (DOI: 10.2140/jsag.2023.13.45). That version can be obtained from the journal or from the Macaulay2 source code repository.


This documentation describes version 2.1 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 a package.


Defining and computing actions

Characters and related operations

Character tables and decompositions

Additional methods