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<head><title>Schubert(ZZ,ZZ,VisibleList) -- find the Pluecker ideal of a Schubert variety</title>
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<div><h1>Schubert(ZZ,ZZ,VisibleList) -- find the Pluecker ideal of a Schubert variety</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>Schubert(k,n,sigma)</tt></div>
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<li><span>Function: <a href="___Schubert_lp__Z__Z_cm__Z__Z_cm__Visible__List_rp.html" title="find the Pluecker ideal of a Schubert variety">Schubert</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>k</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>n</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>sigma</tt>, <span>a <a href="___Visible__List.html">visible list</a></span>, a subset of <tt>0..n</tt> of size <tt>k+1</tt> that indexes the Schubert variety</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, the ideal of the Schubert variety indexed by sigma</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>CoefficientRing => </tt><span><span>a <a href="___Ring.html">ring</a></span>, <span>default value ZZ</span>, the coefficient ring for the polynomial ring to be made</span></span></li>
<li><span><tt>Variable => </tt><span><span>a <a href="___Symbol.html">symbol</a></span>, <span>default value p</span>, the base symbol for the indexed variables to be used. The subscripts are the elements of <tt>subsets(n+1,k+1)</tt>, converted to sequences and, if <tt>k</tt> is 0, converted to integers.</span></span></li>
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<div class="single"><h2>Description</h2>
<div>Given natural numbers <i>k ≤ n</i>, this routine finds the ideal of the Schubert variety indexed by sigma in the Grassmannian of projective <i>k</i>-planes in <i>P<sup>n</sup></i>.<p>For example, the ideal of the Schubert variety indexed by <i>{1,2,4}</i> in the Grassmannian of projective planes in <i>P<sup>4</sup></i> is displayed in the following example.</p>
<table class="examples"><tr><td><pre>i1 : I = Schubert(2,4,{1,2,4},CoefficientRing => QQ)
o1 = ideal (p , p , p , p p - p p , p p
2,3,4 1,3,4 0,3,4 1,2,3 0,2,4 0,2,3 1,2,4 1,2,3 0,1,4
------------------------------------------------------------------------
- p p , p p - p p )
0,1,3 1,2,4 0,2,3 0,1,4 0,1,3 0,2,4
o1 : Ideal of QQ[p , p , p , p , p , p , p , p , p , p ]
0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4</pre>
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<tr><td><pre>i2 : R = ring I;</pre>
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<tr><td><pre>i3 : C = res I
1 6 14 16 9 2
o3 = R <-- R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5 6
o3 : ChainComplex</pre>
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<tr><td><pre>i4 : betti C
0 1 2 3 4 5
o4 = total: 1 6 14 16 9 2
0: 1 3 3 1 . .
1: . 3 11 15 9 2
o4 : BettiTally</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Grassmannian_lp__Z__Z_cm__Z__Z_rp.html" title="the Grassmannian of linear subspaces of a vector space">Grassmannian</a> -- the Grassmannian of linear subspaces of a vector space</span></li>
<li><span><a href="_pfaffians.html" title="ideal generated by Pfaffians">pfaffians</a> -- ideal generated by Pfaffians</span></li>
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