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<head><title>SchurRing -- the class of all Schur rings</title>
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<div><h1>SchurRing -- the class of all Schur rings</h1>
<div class="single"><h2>Description</h2>
<div>A Schur ring is the representation ring for the general linear group of n by n matrices, and one can be constructed with <a href="_schur__Ring.html" title="make a Schur ring">schurRing</a>.<table class="examples"><tr><td><pre>i1 : R = schurRing(s, 4)
o1 = R
o1 : SchurRing</pre>
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The element corresponding to the Young diagram <tt>{3,2,1}</tt> is obtained as follows.<table class="examples"><tr><td><pre>i2 : s_{3,2,1}
o2 = s
3,2,1
o2 : R</pre>
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The dimension of the underlying virtual representation can be obtained with <a href="../../Macaulay2Doc/html/_dim.html" title="compute the Krull dimension">dim</a>.<table class="examples"><tr><td><pre>i3 : dim s_{3,2,1}
o3 = 64</pre>
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Multiplication in the ring comes from tensor product of representations.<table class="examples"><tr><td><pre>i4 : s_{3,2,1} * s_{1,1}
o4 = s + s + s + s + s + s
4,3,1 4,2,2 4,2,1,1 3,3,2 3,3,1,1 3,2,2,1
o4 : R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_schur__Ring.html" title="make a Schur ring">schurRing</a> -- make a Schur ring</span></li>
</ul>
</div>
<div class="waystouse"><h2>Functions and methods returning a Schur ring :</h2>
<ul><li><span>schurRing(Symbol,ZZ), see <span><a href="_schur__Ring.html" title="make a Schur ring">schurRing</a> -- make a Schur ring</span></span></li>
<li><span>schurRing(Thing,ZZ), see <span><a href="_schur__Ring.html" title="make a Schur ring">schurRing</a> -- make a Schur ring</span></span></li>
</ul>
<h2>Methods that use a Schur ring :</h2>
<ul><li>coefficientRing(SchurRing)</li>
<li>degreeLength(SchurRing)</li>
<li>monoid(SchurRing)</li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Schur__Ring.html" title="the class of all Schur rings">SchurRing</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="../../Macaulay2Doc/html/___Engine__Ring.html" title="the class of rings handled by the engine">EngineRing</a> < <a href="../../Macaulay2Doc/html/___Ring.html" title="the class of all rings">Ring</a> < <a href="../../Macaulay2Doc/html/___Type.html" title="the class of all types">Type</a> < <a href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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