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<!DOCTYPE html> <html lang="en"> <head> <title>NOBody -- Newton-Okounkov body of the matching field</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 matching fields for Grassmannians and partial flag varieties" href="index.html">MatchingFields</a> » <a title="Newton-Okounkov body of the matching field" href="___N__O__Body.html">NOBody</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="_weight__Matrix__Cone.html">next</a> | <a href="_matching__Field__Polytope.html">previous</a> | <a href="_weight__Matrix__Cone.html">forward</a> | <a href="___Extra__Zero__Rows.html">backward</a> | <a href="index.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>NOBody -- Newton-Okounkov body of the matching field</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">D = NOBody L</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">L</span>, <span>an instance of the type <a title="the class of Grassmannian matching fields" href="___Gr__Matching__Field.html">GrMatchingField</a></span> or <span>an instance of the type <a title="the class of matching fields for partial flag varieties" href="___Fl__Matching__Field.html">FlMatchingField</a></span>, </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">D</span>, <span>a <a title="the class of all convex polyhedra" href="../../Polyhedra/html/___Polyhedron.html">convex polyhedron</a></span>, Newton-Okounkov body of the matching field</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The Pluecker algebra is generated by Pluecker forms given by top-justified minors of a generic matrix. The Pluecker algebra can be constructed with the function <a title="Pluecker algebra of a (partial) flag variety" href="_pluecker__Algebra.html">plueckerAlgebra</a>, of the image of the Pluecker ring map that can be accessed with the function <a title="The ring map of the Pluecker embedding" href="_pluecker__Map.html">plueckerMap</a>. Note that the ambient ring containing the Pluecker algebra has a weight-based term order that comes from the matching field. We compute a subalgebra basis (SAGBI basis) using the package <a title="A package for finding canonical subalgebra bases (SAGBI bases)" href="../../SubalgebraBases/html/index.html">SubalgebraBases</a> for the Pluecker algebra.</p> <p>The Newton-Okounkov body of the matching field is constructed from this subalgebra basis. In the case of Grassmannian matching fields, the NO body is simply the convex hull of the exponent vectors of the initial terms of the subalgebra basis. If the matching field gives rise to a toric degeneration (see the function <a title="Does the matching field give rise to a toric degeneration" href="_is__Toric__Degeneration.html">isToricDegeneration</a>) then the NO body coincides with the matching field polytope (see <a title="The polytope of a matching field" href="_matching__Field__Polytope.html">matchingFieldPolytope</a>) because the maximal minors form a subalgebra basis for the Pluecker algebra.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : L = diagonalMatchingField(2, 4) o1 = Grassmannian Matching Field for Gr(2, 4) o1 : GrMatchingField</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : P = matchingFieldPolytope L o2 = P o2 : Polyhedron</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : vertices P o3 = | 1 1 0 1 0 0 | | 0 0 1 0 1 0 | | 0 0 0 0 0 1 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 1 0 0 0 0 0 | | 0 1 1 0 0 0 | | 0 0 0 1 1 1 | 8 6 o3 : Matrix QQ <-- QQ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : noBody = NOBody L o4 = noBody o4 : Polyhedron</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : vertices noBody o5 = | 1 1 0 1 0 0 | | 0 0 1 0 1 0 | | 0 0 0 0 0 1 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 1 0 0 0 0 0 | | 0 1 1 0 0 0 | | 0 0 0 1 1 1 | 8 6 o5 : Matrix QQ <-- QQ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : P == noBody o6 = true</code></pre> </td> </tr> </table> <div> <p>In the case of flag matching fields, the NO body is computed in a similar way. First a subalgebra basis is computed for the Pluecker algebra. However, to construct the NO body from the subalgebra basis, we need to take into account the grading on the Pluecker forms. From the geometric perspective, we are simply using the Segre embedding to view the flag variety as a subvariety of a suitably large projective space.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i7 : L = diagonalMatchingField({1,2}, 4) o7 = Flag Matching Field for Fl(1, 2; 4) o7 : FlMatchingField</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : noBody = NOBody L o8 = noBody o8 : Polyhedron</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : vertices noBody o9 = | 2 1 1 1 2 0 1 0 1 0 2 0 0 1 0 0 | | 0 1 0 0 0 2 0 1 0 1 0 2 0 0 1 0 | | 0 0 1 0 0 0 1 1 0 0 0 0 2 0 0 1 | | 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 | 8 16 o9 : Matrix QQ <-- QQ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : noBody == matchingFieldPolytope L o10 = true</code></pre> </td> </tr> </table> <div> <p>Note that the matching field polytope is equal to the NO body if and only if the matching field gives rise to a toric degeneration. So, for a <i>hexagonal matching field</i> for Gr$(3,6)$, the NO body has an additional vertex. We construct a hexagonal matching field using the function <a title="matching field parametrised by permutations" href="_matching__Field__From__Permutation.html">matchingFieldFromPermutation</a> as follows.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i11 : L = matchingFieldFromPermutation(3, 6, {6,1,5,2,3,4}, UsePrimePowers => true, ScalingCoefficient => 3) o11 = Grassmannian Matching Field for Gr(3, 6) o11 : GrMatchingField</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : isToricDegeneration L o12 = false</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : vertices NOBody L o13 = | 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1/2 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 | | 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1/2 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1/2 | | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1/2 | | 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/2 | | 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1/2 | 18 21 o13 : Matrix QQ <-- QQ</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="matching field parametrised by permutations" href="_matching__Field__From__Permutation.html">matchingFieldFromPermutation</a> -- matching field parametrised by permutations</span></li> <li><span><a title="Does the matching field give rise to a toric degeneration" href="_is__Toric__Degeneration.html">isToricDegeneration</a> -- Does the matching field give rise to a toric degeneration</span></li> <li><span><a title="A package for finding canonical subalgebra bases (SAGBI bases)" href="../../SubalgebraBases/html/index.html">SubalgebraBases</a> -- A package for finding canonical subalgebra bases (SAGBI bases)</span></li> <li><span><a title="Pluecker algebra of a (partial) flag variety" href="_pluecker__Algebra.html">plueckerAlgebra</a> -- Pluecker algebra of a (partial) flag variety</span></li> <li><span><a title="The ring map of the Pluecker embedding" href="_pluecker__Map.html">plueckerMap</a> -- The ring map of the Pluecker embedding</span></li> </ul> </div> <div> <h3>Menu</h3> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">NOBody</span>:</h2> <ul> <li><kbd>NOBody(FlMatchingField)</kbd></li> <li><kbd>NOBody(GrMatchingField)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Newton-Okounkov body of the matching field" href="___N__O__Body.html">NOBody</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">MatchingFields.m2:2540:0</span>.</p> </div> </div> </div> </body> </html>
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