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<!DOCTYPE html> <html lang="en"> <head> <title>plueckerPoset -- computes a poset associated to the Plücker relations</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="a package for working with partially ordered sets" href="index.html">Posets</a> :: <a title="computes a poset associated to the Plücker relations" href="_pluecker__Poset.html">plueckerPoset</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="_poincare__Polynomial.html">next</a> | <a href="_partition__Lattice.html">previous</a> | <a href="_poincare__Polynomial.html">forward</a> | <a href="_partition__Lattice.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>plueckerPoset -- computes a poset associated to the Plücker relations</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">P = plueckerPoset n</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">n</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the size of the set to partition</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">P</span>, <span>an instance of the type <a title="a class for partially ordered sets (posets)" href="___Poset.html">Poset</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The ideal of Plücker relations has a quadratic Groebner basis. Under a suitable term order, the incomparable pairs of the poset $P$ generate the initial ideal of the ideal of Plücker relations.</p> <p>Given two subsets $S$ and $T$ of ${0,\ldots,n-1}$, we partially order $S \leq T$ if $\#S \geq \#T$ and $S_i \leq T_i$ for all $i$ from $1$ to $\#T$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : P = plueckerPoset 4;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : coveringRelations P o2 = {{{0}, {1}}, {{1}, {2}}, {{0, 1}, {0, 2}}, {{2}, {3}}, {{0, 2}, {0, 3}}, ------------------------------------------------------------------------ {{0, 2}, {1, 2}}, {{1, 2}, {1, 3}}, {{0, 1, 2}, {0, 1, 3}}, {{3}, {}}, ------------------------------------------------------------------------ {{0, 3}, {0}}, {{0, 3}, {1, 3}}, {{1, 3}, {2, 3}}, {{1, 3}, {1}}, {{0, ------------------------------------------------------------------------ 1, 3}, {0, 2, 3}}, {{0, 1, 3}, {0, 1}}, {{2, 3}, {2}}, {{0, 2, 3}, {0, ------------------------------------------------------------------------ 2}}, {{0, 2, 3}, {1, 2, 3}}, {{1, 2, 3}, {1, 2}}, {{0, 1, 2, 3}, {0, 1, ------------------------------------------------------------------------ 2}}} o2 : List</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">plueckerPoset</span>:</h2> <ul> <li><kbd>plueckerPoset(ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="computes a poset associated to the Plücker relations" href="_pluecker__Poset.html">plueckerPoset</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Posets.m2:3753:0</span>.</p> </div> </div> </div> </body> </html>