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<head><title>SchurRings -- rings representing irreducible representations of GL(n)</title>
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<div><h1>SchurRings -- rings representing irreducible representations of GL(n)</h1>
<div class="single"><h2>Description</h2>
<div>This package make computations in the representation ring of GL(n) possible.<p/>
Given a positive integer <tt>n</tt>, we may define a polynomial ring over <a href="../../Macaulay2Doc/html/___Z__Z.html" title="the class of all integers">ZZ</a> in <tt>n</tt> variables, whose monomials correspond to the irreducible representations of GL(n), and where multiplication is given by the decomposition of the tensor product of representations<p/>
We create such a ring in Macaulay2 using the <a href="_schur__Ring.html" title="make a Schur ring">schurRing</a> function:<table class="examples"><tr><td><pre>i1 : R = schurRing(s,4);</pre>
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A monomial represents the irreducible representation with a given highest weight. The standard 4 dimensional representation is<table class="examples"><tr><td><pre>i2 : V = s_{1}
o2 = s
1
o2 : R</pre>
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We may see the dimension of the corresponding irreducible representation using <a href="../../Macaulay2Doc/html/_dim.html" title="compute the Krull dimension">dim</a>:<table class="examples"><tr><td><pre>i3 : dim V
o3 = 4</pre>
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The third symmetric power of V is obtained by<table class="examples"><tr><td><pre>i4 : W = s_{3}
o4 = s
3
o4 : R</pre>
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<tr><td><pre>i5 : dim W
o5 = 20</pre>
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and the third exterior power of V can be obtained using<table class="examples"><tr><td><pre>i6 : U = s_{1,1,1}
o6 = s
1,1,1
o6 : R</pre>
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<tr><td><pre>i7 : dim U
o7 = 4</pre>
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Multiplication of elements corresponds to tensor product of representations. The value is computed using a variant of the Littlewood-Richardson rule.<table class="examples"><tr><td><pre>i8 : V * V
o8 = s + s
2 1,1
o8 : R</pre>
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<tr><td><pre>i9 : V^3
o9 = s + 2s + s
3 2,1 1,1,1
o9 : R</pre>
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One cannot make quotients of this ring, and Groebner bases and related computations do not work, but I'm not sure what they would mean...</div>
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<div class="single"><h2>Authors</h2>
<ul><li><div class="single"><a href="http://www.math.cornell.edu/~mike/">Michael Stillman</a><span> <<a href="mailto:mike@math.cornell.edu">mike@math.cornell.edu</a>></span></div>
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<li><div class="single">Hal Schenck</div>
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<div class="single"><h2>Version</h2>
This documentation describes version <b>0.2</b> of SchurRings.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/SchurRings.m2">SchurRings.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<ul><li><span><a href="___Schur__Ring.html" title="the class of all Schur rings">SchurRing</a> -- the class of all Schur rings</span></li>
<li><span><a href="___Schur__Ring__Indexed__Variable__Table.html" title="">SchurRingIndexedVariableTable</a></span></li>
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<li><div class="single">Functions<ul><li><span><tt>jacobiTrudi</tt> (missing documentation<!-- tag: jacobiTrudi -->)</span></li>
<li><span><a href="_schur__Ring.html" title="make a Schur ring">schurRing</a> -- make a Schur ring</span></li>
<li><span><tt>symmRing</tt> (missing documentation<!-- tag: symmRing -->)</span></li>
<li><span><tt>toE</tt> (missing documentation<!-- tag: toE -->)</span></li>
<li><span><tt>toP</tt> (missing documentation<!-- tag: toP -->)</span></li>
<li><span><tt>toS</tt> (missing documentation<!-- tag: toS -->)</span></li>
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<li><div class="single">Methods<ul><li><span>SchurRingIndexedVariableTable _ Thing, see <span><a href="___Schur__Ring__Indexed__Variable__Table.html" title="">SchurRingIndexedVariableTable</a></span></span></li>
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