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<!DOCTYPE html> <html lang="en"> <head> <title>torusInvariants -- ring of invariants of torus action</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="an interface to use Normaliz in Macaulay 2" href="index.html">Normaliz</a> :: <a title="ring of invariants of torus action" href="_torus__Invariants.html">torusInvariants</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> <a href="_vars_lp__Monomial__Subalgebra_rp.html">next</a> | <a href="___The_splattice_sppoints_cm_sp__Ehrhart_spseries_spand_spthe_spsupport_sphyperplanes_spof_span_spintegral_sppolytope..html">previous</a> | <a href="_vars_lp__Monomial__Subalgebra_rp.html">forward</a> | <a href="___The_splattice_sppoints_cm_sp__Ehrhart_spseries_spand_spthe_spsupport_sphyperplanes_spof_span_spintegral_sppolytope..html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>torusInvariants -- ring of invariants of torus action</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">torusInvariants(T,R)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, matrix (a_{ij}) of the action</span></li> <li><span><span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, the ring on which the action takes place</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">allComputations</span> (missing documentation)<!--tag: [torusInvariants, allComputations]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span></span></span></li> <li><span><span class="tt">grading</span> (missing documentation)<!--tag: [torusInvariants, grading]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span></span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="class of monomial subalgebras" href="___Monomial__Subalgebra.html">monomial subalgebra</a></span>, the ring of invariants R^T</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <p></p> Let T=(K^*)^r be the r-dimensional torus acting on the polynomial ring R=K[X_1,\ldots,X_n] diagonally. Such an action can be described as follows: there are integers a_{ij}, i=1,\ldots,r, j=1,\ldots,n, such that (\lambda_1,\ldots,\lambda_r)\in T acts by the substitution <br><br>X_j\mapsto \lambda_1^{a_{1j}}*\ldots*\lambda_r^{a_{rj}}X_j, j=1,\ldots,n.<br><br>The function takes the matrix (a_{ij}) as input and computes the ring of invariants R^T=\{f\in R: \lambda f=f for all \lambda \in T\}. <p></p> This method can be used with the options <a href="_all__Computations.html">allComputations</a> and <a href="_grading.html">grading</a>. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R=QQ[x,y,z,w];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : T=matrix({{-1,-1,2,0},{1,1,-2,-1}}); 2 4 o2 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : torusInvariants(T,R) o3 = MonomialSubalgebra{cache => CacheTable{...1...} } 2 2 generators => {y z, x*y*z, x z} ring => R o3 : MonomialSubalgebra of R</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="ring of invariants of a finite group action" href="_finite__Diag__Invariants.html">finiteDiagInvariants</a> -- ring of invariants of a finite group action</span></li> <li><span><a title="ring of invariants of a diagonalizable group action" href="_diag__Invariants.html">diagInvariants</a> -- ring of invariants of a diagonalizable group action</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">torusInvariants</span>:</h2> <ul> <li><kbd>torusInvariants(Matrix,Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="ring of invariants of torus action" href="_torus__Invariants.html">torusInvariants</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Normaliz.m2:1932:0</span>.</p> </div> </div> </div> </body> </html>