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<!DOCTYPE html> <html lang="en"> <head> <title>BettiCharacters -- finite group characters on free resolutions and graded modules</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="finite group characters on free resolutions and graded modules" href="index.html">BettiCharacters</a> :: <a title="finite group characters on free resolutions and graded modules" href="index.html">BettiCharacters</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> next | previous | <a href="_action.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>BettiCharacters -- finite group characters on free resolutions and graded modules</h1> <div> <h2>Description</h2> <div> <p>This package contains functions for computing characters of free resolutions and graded modules equipped with the action of a finite group.</p> <p>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 <b>Betti characters</b> 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.</p> <p>This package provides functions to compute the Betti characters and the characters of graded components of $M$ based on the algorithms in <a href="https://doi.org/10.1016/j.jsc.2022.02.001">F. Galetto - Finite group characters on free resolutions</a>. 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.</p> <h4>Version history:</h4> <ul> <li><b>1.0: </b>Initial version. Includes computation of actions and Betti characters.</li> <li><b>2.0: </b>Introduces character tables, decompositions, and other methods for characters.</li> <li><b>2.1: </b>Adds equality checks for actions and characters. Contains several small improvements to the code and documentation, including a new multigraded example.</li> </ul> </div> </div> <div> <h3>Menu</h3> <h4>Defining and computing actions</h4> <ul> <li><span><a title="define finite group action" href="_action.html">action</a> -- define finite group action</span></li> <li><span><a title="group elements of an action" href="_actors.html">actors</a> -- group elements of an action</span></li> </ul> <h4>Characters and related operations</h4> <ul> <li><span><a title="compute characters of finite group action" href="_character.html">character</a> -- compute characters of finite group action</span></li> <li><span><a title="shift, direct sum, dual, and tensor product" href="___Character_spoperations.html">Character operations</a> -- shift, direct sum, dual, and tensor product</span></li> </ul> <h4>Character tables and decompositions</h4> <ul> <li><span><a title="construct a character table" href="_character__Table.html">characterTable</a> -- construct a character table</span></li> <li><span><a title="decompose a character into irreducible characters" href="_decompose__Character.html">decomposeCharacter</a> -- decompose a character into irreducible characters</span></li> </ul> <h4>Additional methods</h4> <ul> <li><span><a title="compare actions and characters" href="___Equality_spchecks.html">Equality checks</a> -- compare actions and characters</span></li> <li><span><a title="permutation action of the symmetric group" href="_symmetric__Group__Actors.html">symmetricGroupActors</a> -- permutation action of the symmetric group</span></li> <li><span><a title="character table of the symmetric group" href="_symmetric__Group__Table.html">symmetricGroupTable</a> -- character table of the symmetric group</span></li> </ul> <h4>Examples</h4> <ul> <li><span><a title="Specht ideals / subspace arrangements" href="___Betti__Characters_sp__Example_sp1.html">BettiCharacters Example 1</a> -- Specht ideals / subspace arrangements</span></li> <li><span><a title="Symbolic powers of star configurations" href="___Betti__Characters_sp__Example_sp2.html">BettiCharacters Example 2</a> -- Symbolic powers of star configurations</span></li> <li><span><a title="Klein configuration of points" href="___Betti__Characters_sp__Example_sp3.html">BettiCharacters Example 3</a> -- Klein configuration of points</span></li> <li><span><a title="a multigraded example" href="___Betti__Characters_sp__Example_sp4.html">BettiCharacters Example 4</a> -- a multigraded example</span></li> </ul> </div> <div> <div> <div> <h2>Author</h2> <ul> <li><a href="http://math.galetto.org">Federico Galetto</a><span> <<a href="mailto:galetto.federico%40gmail.com">galetto.federico@gmail.com</a>></span></li> </ul> </div> <div> <h2>Certification <img src="../../../../Macaulay2/Style/GoldStar.png" alt="a gold star"> </h2> <p>Version <b>2.1</b> of this package was accepted for publication in <a href="https://msp.org/jsag/2023/13-1/">volume 13</a> of <a href="https://msp.org/jsag/">Journal of Software for Algebra and Geometry</a> on 2023-05-30, in the article <a href="https://msp.org/jsag/2023/13-1/p04.xhtml">Setting the scene for Betti characters</a> (DOI: <a href="https://doi.org/10.2140/jsag.2023.13.45">10.2140/jsag.2023.13.45</a>). That version can be obtained from <a href="https://msp.org/jsag/2023/13-1/jsag-v13-n1-x04-BettiCharacters.m2">the journal</a>.</p> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>2.1</b> of BettiCharacters, released <b>February 26, 2023</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{BettiCharactersSource, title = {{BettiCharacters: finite group characters on free resolutions and graded modules. Version~2.1}}, author = {Federico Galetto}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } @article{BettiCharactersArticle, title = {{Setting the scene for Betti characters}}, author = {Federico Galetto}, journal = {Journal of Software for Algebra and Geometry}, volume = {13}, year = {2023}, } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Types <ul> <li><span><a title="the class of all finite group actions" href="___Action.html">Action</a> -- the class of all finite group actions</span></li> <li><span><a title="the class of all finite group actions on complexes" href="___Action__On__Complex.html">ActionOnComplex</a> -- the class of all finite group actions on complexes</span></li> <li><span><a title="the class of all finite group actions on graded modules" href="___Action__On__Graded__Module.html">ActionOnGradedModule</a> -- the class of all finite group actions on graded modules</span></li> <li><span><a title="the class of all characters of finite group representations" href="___Character.html">Character</a> -- the class of all characters of finite group representations</span></li> <li><span><a title="the class of all finite group character decompositions" href="___Character__Decomposition.html">CharacterDecomposition</a> -- the class of all finite group character decompositions</span></li> <li><span><a title="the class of all character tables of finite groups" href="___Character__Table.html">CharacterTable</a> -- the class of all character tables of finite groups</span></li> </ul> </li> <li>Functions and commands <ul> <li><span><a title="define finite group action" href="_action.html">action</a> -- define finite group action</span></li> <li><span><a title="group elements of an action" href="_actors.html">actors</a> -- group elements of an action</span></li> <li><span><a title="compute characters of finite group action" href="_character.html">character</a> -- compute characters of finite group action</span></li> <li><span><a title="construct a character table" href="_character__Table.html">characterTable</a> -- construct a character table</span></li> <li><span><a title="decompose a character into irreducible characters" href="_decompose__Character.html">decomposeCharacter</a> -- decompose a character into irreducible characters</span></li> <li><span><a title="get inverse of action on ring generators" href="_inverse__Ring__Actors.html">inverseRingActors</a> -- get inverse of action on ring generators</span></li> <li><span><a title="number of acting elements" href="_num__Actors.html">numActors</a> -- number of acting elements</span></li> <li><span><a title="get action on ring generators" href="_ring__Actors.html">ringActors</a> -- get action on ring generators</span></li> <li><span><a title="permutation action of the symmetric group" href="_symmetric__Group__Actors.html">symmetricGroupActors</a> -- permutation action of the symmetric group</span></li> <li><span><a title="character table of the symmetric group" href="_symmetric__Group__Table.html">symmetricGroupTable</a> -- character table of the symmetric group</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>action(Complex,List)</kbd> -- see <span><a title="define finite group action on a resolution" href="_action_lp__Complex_cm__List_cm__List_cm__Z__Z_rp.html">action(Complex,List,List,ZZ)</a> -- define finite group action on a resolution</span></span></li> <li><span><a title="define finite group action on a resolution" href="_action_lp__Complex_cm__List_cm__List_cm__Z__Z_rp.html">action(Complex,List,List,ZZ)</a> -- define finite group action on a resolution</span></li> <li><span><kbd>action(Ideal,List)</kbd> -- see <span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></span></li> <li><span><kbd>action(Ideal,List,List)</kbd> -- see <span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></span></li> <li><span><kbd>action(Module,List)</kbd> -- see <span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></span></li> <li><span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></li> <li><span><kbd>action(PolynomialRing,List)</kbd> -- see <span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></span></li> <li><span><kbd>action(PolynomialRing,List,List)</kbd> -- see <span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></span></li> <li><span><kbd>action(QuotientRing,List)</kbd> -- see <span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></span></li> <li><span><kbd>action(QuotientRing,List,List)</kbd> -- see <span><a title="define finite group action on a graded module" href="_action_lp__Module_cm__List_cm__List_rp.html">action(Module,List,List)</a> -- define finite group action on a graded module</span></span></li> <li><span><a title="group elements of action on resolution" href="_actors_lp__Action__On__Complex_cm__Z__Z_rp.html">actors(ActionOnComplex,ZZ)</a> -- group elements of action on resolution</span></li> <li><span><a title="group elements acting on components of a module" href="_actors_lp__Action__On__Graded__Module_cm__List_rp.html">actors(ActionOnGradedModule,List)</a> -- group elements acting on components of a module</span></li> <li><span><kbd>actors(ActionOnGradedModule,ZZ)</kbd> -- see <span><a title="group elements acting on components of a module" href="_actors_lp__Action__On__Graded__Module_cm__List_rp.html">actors(ActionOnGradedModule,List)</a> -- group elements acting on components of a module</span></span></li> <li><span><a title="homological shift" href="___Character_sp__Array.html">Character Array</a> -- homological shift</span></li> <li><span><a title="compute all Betti characters of minimal free resolution" href="_character_lp__Action__On__Complex_rp.html">character(ActionOnComplex)</a> -- compute all Betti characters of minimal free resolution</span></li> <li><span><a title="compute Betti characters of minimal free resolution" href="_character_lp__Action__On__Complex_cm__Z__Z_rp.html">character(ActionOnComplex,ZZ)</a> -- compute Betti characters of minimal free resolution</span></li> <li><span><a title="compute characters of graded components of a module" href="_character_lp__Action__On__Graded__Module_cm__List_rp.html">character(ActionOnGradedModule,List)</a> -- compute characters of graded components of a module</span></li> <li><span><kbd>character(ActionOnGradedModule,ZZ)</kbd> -- see <span><a title="compute characters of graded components of a module" href="_character_lp__Action__On__Graded__Module_cm__List_rp.html">character(ActionOnGradedModule,List)</a> -- compute characters of graded components of a module</span></span></li> <li><span><kbd>character(ActionOnGradedModule,ZZ,ZZ)</kbd> -- see <span><a title="compute characters of graded components of a module" href="_character_lp__Action__On__Graded__Module_cm__List_rp.html">character(ActionOnGradedModule,List)</a> -- compute characters of graded components of a module</span></span></li> <li><span><a title="recover character from decomposition" href="_character_lp__Character__Decomposition_cm__Character__Table_rp.html">character(CharacterDecomposition,CharacterTable)</a> -- recover character from decomposition</span></li> <li><span><kbd>CharacterDecomposition * CharacterTable</kbd> -- see <span><a title="recover character from decomposition" href="_character_lp__Character__Decomposition_cm__Character__Table_rp.html">character(CharacterDecomposition,CharacterTable)</a> -- recover character from decomposition</span></span></li> <li><span><a title="construct a character" href="_character_lp__Polynomial__Ring_cm__Z__Z_cm__Hash__Table_rp.html">character(PolynomialRing,ZZ,HashTable)</a> -- construct a character</span></li> <li><span><kbd>characterTable(List,Matrix,PolynomialRing,List)</kbd> -- see <span><a title="construct a character table" href="_character__Table.html">characterTable</a> -- construct a character table</span></span></li> <li><span><kbd>characterTable(List,Matrix,PolynomialRing,RingMap)</kbd> -- see <span><a title="construct a character table" href="_character__Table.html">characterTable</a> -- construct a character table</span></span></li> <li><span><kbd>Character / CharacterTable</kbd> -- see <span><a title="decompose a character into irreducible characters" href="_decompose__Character.html">decomposeCharacter</a> -- decompose a character into irreducible characters</span></span></li> <li><span><kbd>decomposeCharacter(Character,CharacterTable)</kbd> -- see <span><a title="decompose a character into irreducible characters" href="_decompose__Character.html">decomposeCharacter</a> -- decompose a character into irreducible characters</span></span></li> <li><span><kbd>Character ++ Character</kbd> -- see <span><a title="direct sum of characters" href="_direct__Sum_lp__Character_rp.html">directSum(Character)</a> -- direct sum of characters</span></span></li> <li><span><a title="direct sum of characters" href="_direct__Sum_lp__Character_rp.html">directSum(Character)</a> -- direct sum of characters</span></li> <li><span><kbd>dual(Character,List)</kbd> -- see <span><a title="dual character" href="_dual.html">dual</a> -- dual character</span></span></li> <li><span><kbd>dual(Character,RingMap)</kbd> -- see <span><a title="dual character" href="_dual.html">dual</a> -- dual character</span></span></li> <li><span><kbd>ActionOnComplex == ActionOnComplex</kbd> -- see <span><a title="compare actions and characters" href="___Equality_spchecks.html">Equality checks</a> -- compare actions and characters</span></span></li> <li><span><kbd>ActionOnGradedModule == ActionOnGradedModule</kbd> -- see <span><a title="compare actions and characters" href="___Equality_spchecks.html">Equality checks</a> -- compare actions and characters</span></span></li> <li><span><kbd>Character == Character</kbd> -- see <span><a title="compare actions and characters" href="___Equality_spchecks.html">Equality checks</a> -- compare actions and characters</span></span></li> <li><span><kbd>inverseRingActors(Action)</kbd> -- see <span><a title="get inverse of action on ring generators" href="_inverse__Ring__Actors.html">inverseRingActors</a> -- get inverse of action on ring generators</span></span></li> <li><span><a title="format for printing, as a net" href="_net_lp__Action_rp.html">net(Action)</a> -- format for printing, as a net</span></li> <li><span><a title="format for printing, as a net" href="_net_lp__Character_rp.html">net(Character)</a> -- format for printing, as a net</span></li> <li><span><a title="format for printing, as a net" href="_net_lp__Character__Decomposition_rp.html">net(CharacterDecomposition)</a> -- format for printing, as a net</span></li> <li><span><a title="format for printing, as a net" href="_net_lp__Character__Table_rp.html">net(CharacterTable)</a> -- format for printing, as a net</span></li> <li><span><kbd>numActors(Action)</kbd> -- see <span><a title="number of acting elements" href="_num__Actors.html">numActors</a> -- number of acting elements</span></span></li> <li><span><a title="get ring of object acted upon" href="_ring_lp__Action_rp.html">ring(Action)</a> -- get ring of object acted upon</span></li> <li><span><kbd>ringActors(Action)</kbd> -- see <span><a title="get action on ring generators" href="_ring__Actors.html">ringActors</a> -- get action on ring generators</span></span></li> <li><span><kbd>symmetricGroupActors(PolynomialRing)</kbd> -- see <span><a title="permutation action of the symmetric group" href="_symmetric__Group__Actors.html">symmetricGroupActors</a> -- permutation action of the symmetric group</span></span></li> <li><span><kbd>symmetricGroupTable(PolynomialRing)</kbd> -- see <span><a title="character table of the symmetric group" href="_symmetric__Group__Table.html">symmetricGroupTable</a> -- character table of the symmetric group</span></span></li> <li><span><a title="get object acted upon" href="_target_lp__Action_rp.html">target(Action)</a> -- get object acted upon</span></li> <li><span><kbd>Character ** Character</kbd> -- see <span><a title="tensor product of characters" href="_tensor_lp__Character_cm__Character_rp.html">tensor(Character,Character)</a> -- tensor product of characters</span></span></li> <li><span><a title="tensor product of characters" href="_tensor_lp__Character_cm__Character_rp.html">tensor(Character,Character)</a> -- tensor product of characters</span></li> </ul> </li> <li>Symbols <ul> <li><span><a title="custom labels for irreducible characters" href="___Labels.html">Labels</a> -- custom labels for irreducible characters</span></li> <li><span><a title="format ring actors as one-row substitution matrices" href="___Sub.html">Sub</a> -- format ring actors as one-row substitution matrices</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="finite group characters on free resolutions and graded modules" href="index.html">BettiCharacters</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">BettiCharacters.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BettiCharacters.m2:2846:0</span>.</p> </div> </div> </div> </body> </html>