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<head><title>schur -- creates a map between Schur modules</title>
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<div><h1>schur -- creates a map between Schur modules</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>schur(lambda,f)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>lambda</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of numbers representing a partition</span></li>
<li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a map between two free modules</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>F</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the result of applying the Schur functor associated to lambda to f</span></li>
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<div class="single"><h2>Description</h2>
<div><div>Applies the Schur functor associated to lambda to the map f between free modules. The modules <tt>source F</tt> and <tt>target F</tt> are Schur modules containing certain data in cache (see <a href="_schur__Module.html" title="creates Schur module from a partition and free module">schurModule</a>).</div>
<table class="examples"><tr><td><pre>i1 : R=QQ[x_1,x_2,x_3]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : F=map(R^1,R^3,vars R)
o2 = | x_1 x_2 x_3 |
1 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : L=schur({2},F) -- 2nd veronese embedding
o3 = | x_1^2 x_1x_2 x_1x_3 x_2^2 x_2x_3 x_3^2 |
1 6
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : F=matrix{{1_QQ,2,4},{3,9,27},{4,16,64}}
o4 = | 1 2 4 |
| 3 9 27 |
| 4 16 64 |
3 3
o4 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i5 : schur({1,1},F)
o5 = | 3 15 18 |
| 8 48 64 |
| 12 84 144 |
3 3
o5 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i6 : minors(2,F)
o6 = ideal (3, 8, 12, 15, 48, 84, 18, 64, 144)
o6 : Ideal of QQ</pre>
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<tr><td><pre>i7 : schur({1,1,1},F) == det F
o7 = true</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>The partition lambda should be a valid nonempty partition.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_schur__Module.html" title="creates Schur module from a partition and free module">schurModule</a> -- creates Schur module from a partition and free module</span></li>
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<div class="waystouse"><h2>Ways to use <tt>schur</tt> :</h2>
<ul><li>schur(List,Matrix)</li>
</ul>
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