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<head><title>symmetricAlgebra -- the symmetric algebra of a module</title>
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<div><h1>symmetricAlgebra -- the symmetric algebra of a module</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_symmetric__Algebra.html">DegreeLift => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">DegreeMap => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">DegreeRank => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Degrees => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Global => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Heft => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Inverses => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Join => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Local => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">MonomialOrder => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">MonomialSize => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">SkewCommutative => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">VariableBaseName => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Variables => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">Weights => ...</a>, -- the symmetric algebra of a module</span></li>
<li><span><a href="_symmetric__Algebra.html">WeylAlgebra => ...</a>, -- the symmetric algebra of a module</span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>symmetricAlgebra M</tt></div>
</dd></dl>
</div>
</li>
<li>Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
</ul>
</li>
<li>Outputs:<ul><li><span><span>a <a href="___Ring.html">ring</a></span>, the symmetric algebra of <tt>M</tt></span></li>
</ul>
</li>
</ul>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : M = image matrix{{a,b,c}}
o2 = image | a b c |
1
o2 : R-module, submodule of R</pre>
</td></tr>
<tr><td><pre>i3 : symmetricAlgebra M
R[p , p , p ]
0 1 2
o3 = ---------------------------------------------
(- b*p + a*p , - c*p + b*p , - c*p + a*p )
0 1 1 2 0 2
o3 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i4 : symmetricAlgebra(R^{1,2,3})
o4 = R[p , p , p ]
0 1 2
o4 : PolynomialRing</pre>
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</table>
<p>Most of the optional arguments for monoids (see <a href="___Ring_sp__Array.html" title="the standard way to make a polynomial ring">Ring Array</a> or <a href="_monoid.html" title="make or retrieve a monoid">monoid</a>) are available here as well, as in the following example. They apply to the variables that correspond to the generators of the module.</p>
<table class="examples"><tr><td><pre>i5 : A = symmetricAlgebra(M, Variables=>{x,y,z})
o5 = A
o5 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i6 : describe A
R[x, y, z]
o6 = ---------------------------------------
(- b*x + a*y, - c*y + b*z, - c*x + a*z)</pre>
</td></tr>
<tr><td><pre>i7 : B = symmetricAlgebra(M, VariableBaseName=>G, MonomialSize=>16)
o7 = B
o7 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i8 : describe B
R[G , G , G ]
0 1 2
o8 = ---------------------------------------------
(- b*G + a*G , - c*G + b*G , - c*G + a*G )
0 1 1 2 0 2</pre>
</td></tr>
<tr><td><pre>i9 : symmetricAlgebra(M, Degrees=> {3:1})
R[p , p , p ]
0 1 2
o9 = ---------------------------------------------
(- b*p + a*p , - c*p + b*p , - c*p + a*p )
0 1 1 2 0 2
o9 : QuotientRing</pre>
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<div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>symmetricAlgebra f</tt></div>
</dd></dl>
</div>
</li>
<li>Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span></span></li>
</ul>
</li>
<li>Outputs:<ul><li><span><span>a <a href="___Ring__Map.html">ring map</a></span>, the map between symmetric algebras induced by <tt>f</tt></span></li>
</ul>
</li>
</ul>
<table class="examples"><tr><td><pre>i10 : symmetricAlgebra vars R
o10 = map(R[p ],R[p , p , p , p ],{a*p , b*p , c*p , d*p , a, b, c, d})
0 0 1 2 3 0 0 0 0
o10 : RingMap R[p ] <--- R[p , p , p , p ]
0 0 1 2 3</pre>
</td></tr>
</table>
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<div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>symmetricAlgebra(A,B,f)</tt></div>
</dd></dl>
</div>
</li>
<li>Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="___Ring.html">ring</a></span>, the symmetric algebra of the target of <tt>f</tt>, previously computed. Optional.</span></li>
<li><span><tt>B</tt>, <span>a <a href="___Ring.html">ring</a></span>, the symmetric algebra of the source of <tt>f</tt>, previously computed. Optional.</span></li>
<li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span></span></li>
</ul>
</li>
<li>Outputs:<ul><li><span><span>a <a href="___Ring__Map.html">ring map</a></span>, the map between the specified symmetric algebras induced by <tt>f</tt></span></li>
</ul>
</li>
</ul>
<p>This form of funtoriality is needed, because various options are available when computing symmetric algebras.</p>
<p>For a linear map that is an isomorphism, and is known to be so, e.g., by having had its inverse computed, the inverse of the corresponding map between symmetric algebras is precomputed and made available.</p>
<table class="examples"><tr><td><pre>i11 : symmetricAlgebra vars R
o11 = map(R[p ],R[p , p , p , p ],{a*p , b*p , c*p , d*p , a, b, c, d})
0 0 1 2 3 0 0 0 0
o11 : RingMap R[p ] <--- R[p , p , p , p ]
0 0 1 2 3</pre>
</td></tr>
<tr><td><pre>i12 : p = symmetricAlgebra(A,B,id_M)
o12 = map(A,B,{x, y, z, a, b, c, d})
o12 : RingMap A <--- B</pre>
</td></tr>
<tr><td><pre>i13 : p^-1
o13 = map(B,A,{G , G , G , a, b, c, d})
0 1 2
o13 : RingMap B <--- A</pre>
</td></tr>
<tr><td><pre>i14 : p * p^-1 === id_A
o14 = true</pre>
</td></tr>
<tr><td><pre>i15 : p^-1 * p === id_B
o15 = true</pre>
</td></tr>
</table>
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<div class="waystouse"><h2>Ways to use <tt>symmetricAlgebra</tt> :</h2>
<ul><li>symmetricAlgebra(Matrix)</li>
<li>symmetricAlgebra(Module)</li>
<li>symmetricAlgebra(Nothing,Nothing,Matrix)</li>
<li>symmetricAlgebra(Nothing,Ring,Matrix)</li>
<li>symmetricAlgebra(Ring,Nothing,Matrix)</li>
<li>symmetricAlgebra(Ring,Ring,Matrix)</li>
</ul>
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