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<head><title>schurModule -- creates Schur module from a partition and free module</title>
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<div><h1>schurModule -- creates Schur module from a partition and free module</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>schurModule(lambda,E)</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; e.g. 3,1 stands for 2 rows of length 3 and 1.</span></li>
<li><span><tt>E</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a free module</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, The result of application of the Schur functor associated to lambda to E.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Applies the Schur functor associated to lambda to the free module E. For a detailed definition of the Schur module see p.106 of Fulton "Young Tableaux".</p>
<p>The resulting M comes with cached data M.cache.Schur = f, finv, AT, ST where</p>
<p>"f is a map from exterior<sub>m</sub>u E to M;", "finv is a map from M to exterior<sub>m</sub>u E;", "AT is a hash table of all tableaux, whose entries increase in every column;", "ST is a hash table of all standard tableaux (tableaux in AT, whose entries nondecrease in every row)." </p>
<div>Tableaux are represented with objects of class Filling, which is a doble list whose entries are lists giving the fillings of the corresponding columns.</div>
<table class="examples"><tr><td><pre>i1 : M=QQ^3;</pre>
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<tr><td><pre>i2 : scan(4, i-> << i+1 << "-th symmetric power of M = " << schurModule({i+1},M) << endl)
3
1-th symmetric power of M = QQ
6
2-th symmetric power of M = QQ
10
3-th symmetric power of M = QQ
15
4-th symmetric power of M = QQ</pre>
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<tr><td><pre>i3 : S = schurModule({3,2,1}, M);</pre>
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<tr><td><pre>i4 : v = sum(numgens S, i-> (i+1)*S_i) -- an element of S represented by a vector
o4 = | 1 |
| 2 |
| 3 |
| 4 |
| 5 |
| 6 |
| 7 |
| 8 |
8
o4 : QQ</pre>
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<tr><td><pre>i5 : printSchurModuleElement(v, S);
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
1*|0|0|0| 2*|0|0|1| 3*|0|0|2| 4*|0|0|0| 5*|0|0|1| 6*|0|0|2| 7*|0|1|1| 8*|0|1|2|
|1|1| | |1|1| | |1|1| | |1|2| | |1|2| | |1|2| | |1|2| | |1|2| |
|2| | | |2| | | |2| | | |2| | | |2| | | |2| | | |2| | | |2| | |
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+</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.html" title="creates a map between Schur modules">schur</a> -- creates a map between Schur modules</span></li>
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<div class="waystouse"><h2>Ways to use <tt>schurModule</tt> :</h2>
<ul><li>schurModule(List,Module)</li>
</ul>
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