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<!DOCTYPE html> <html lang="en"> <head> <title>garnirElement -- a SpechtModuleElement that is equal to zero</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 constructing Specht Modules" href="index.html">SpechtModule</a> :: <a title="a SpechtModuleElement that is equal to zero" href="_garnir__Element.html">garnirElement</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="_generalized__Vandermonde__Matrix_lp__List_cm__List_cm__Polynomial__Ring_rp.html">next</a> | <a href="_first__Row__Descent_lp__Young__Tableau_rp.html">previous</a> | <a href="_generalized__Vandermonde__Matrix_lp__List_cm__List_cm__Polynomial__Ring_rp.html">forward</a> | <a href="_first__Row__Descent_lp__Young__Tableau_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>garnirElement -- a SpechtModuleElement that is equal to zero</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">garnirElement(y,coef,a,b)</code></dd> <dd><code class="language-macaulay2">garnirElement(y,coef)</code></dd> <dd><code class="language-macaulay2">garnirElement(y)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">y</span>, <span>an instance of the type <a title="the class of Young Tableaux" href="___Young__Tableau.html">YoungTableau</a></span>, a tableau that labels a polytabloid</span></li> <li><span><span class="tt">a</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the row of the descent</span></li> <li><span><span class="tt">b</span>, <span>a <a title="the class of all rational numbers" href="../../Macaulay2Doc/html/___Q__Q.html">rational number</a></span>, the column of the descent</span></li> <li><span><span class="tt">coef</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the coefficient of the polytabloid</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>an instance of the type <a title="the class of Specht Module elements" href="___Specht__Module__Element.html">SpechtModuleElement</a></span>, an element which is equal to zero.</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>A Garnir element is an element which is constructed to remove row descents from a tableau. Given a tableau $T$, the Garnir element is defined for a subset $A$ of the $i$th column and a subset $B$ of the $i+1$ column. It is defined as $ \sum_{\pi} sgn(\pi)\pi(T)$. The $\pi$ are called transversals. They are a set of permutations such that $S_{A \cup B}$ is the disjoint union of $\pi(S_A \times S_B)$.</p> <p>The identity can always be chosen as a transversal for any pair of sets. Therefore the original tableau $T$ appears along side other tableaux which are closer to being standard. Another property is that this element is equal to zero. Therefore the original polytabloid $e_T$ can be written as $ e_T = -\sum_{\pi \neq id} sgn(\pi)\pi(e_T) $</p> <p>In this implementation the $i$th column is taken to be the parameter b. The set $A$ is all the cells in the $i$th column from the a-th row to the bottom. The set $B$ is all the cells in the $i+1$ column from the a-th row to the top.</p> <p>If the number (a,b) are not specified then they are taken as the coordinates of the first row descent of $T$</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : p = new Partition from {3,2,1} o1 = Partition{3, 2, 1} o1 : Partition</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : y = youngTableau(p,{1,2,3,5,4,6}) o2 = | 1 2 3 | | 5 4 | | 6 | o2 : YoungTableau</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : garnirElement y o3 = - | 1 2 3 | + | 1 2 3 | + | 1 4 3 | - | 1 4 3 | + | 1 5 3 | + | 1 2 3 | | 4 5 | | 4 6 | | 2 5 | | 2 6 | | 2 6 | | 5 4 | | 6 | | 5 | | 6 | | 5 | | 4 | | 6 | o3 : SpechtModuleElement</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="retrieves the first row descent of a Young tableau" href="_first__Row__Descent_lp__Young__Tableau_rp.html">firstRowDescent</a> -- retrieves the first row descent of a Young tableau</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">garnirElement</span>:</h2> <ul> <li><kbd>garnirElement(YoungTableau)</kbd></li> <li><kbd>garnirElement(YoungTableau,ZZ)</kbd></li> <li><kbd>garnirElement(YoungTableau,ZZ,ZZ,ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a SpechtModuleElement that is equal to zero" href="_garnir__Element.html">garnirElement</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">SpechtModule.m2:3305:0</span>.</p> </div> </div> </div> </body> </html>