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<head><title>Symmetries -- permutations leaving an ideal invariant to speed up gfan computations.</title>
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<div><h1>Symmetries -- permutations leaving an ideal invariant to speed up gfan computations.</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>Symmetries => L</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of permutations of the variables leaving an ideal invariant.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Many gfan functions can be sped up and give smaller output when symmetries of the ideal are given. Permuations are specified as sequences of variables. Not all permuations need to be listed; only permutations that generate all of the symmetries.</p>
<p>A possible caveat is that the permuatations must be appled to the output to get the entire result.</p>
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<table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c];</pre>
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<tr><td><pre>i2 : I = ideal(a+b,b+c);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : gfan(I)
LP algorithm being used: "cddgmp".
o3 = ({{b, a}, {c, a}, {c, b}}, {{b + c, a - c}, {b + c, a + b}, {- a + c, a
------------------------------------------------------------------------
+ b}})
o3 : Sequence</pre>
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<tr><td><pre>i4 : gfan(I, Symmetries => {(c,b,a)})
LP algorithm being used: "cddgmp".
o4 = ({{b, a}, {c, a}}, {{b + c, a - c}, {b + c, a + b}})
o4 : Sequence</pre>
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<p>Note that the use of symmetries above reduces the amount of output. The permutations must be appled to the output to get the complete result.</p>
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<table class="examples"><tr><td><pre>i5 : R = QQ[a,b,c,d,e];</pre>
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<tr><td><pre>i6 : I = ideal"a+b+c+d,ab+bc+cd+da,abc+bcd+cda+dab,abcd-e4"
o6 = ideal (a + b + c + d, a*b + b*c + a*d + c*d, a*b*c + a*b*d + a*c*d +
------------------------------------------------------------------------
4
b*c*d, a*b*c*d - e )
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : #universalGroebnerBasis(I)
LP algorithm being used: "cddgmp".
o7 = 75</pre>
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<tr><td><pre>i8 : #universalGroebnerBasis(I, Symmetries => {(b,c,d,a,e)})
LP algorithm being used: "cddgmp".
o8 = 27</pre>
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<div class="single"><h2>Functions with optional argument named Symmetries :</h2>
<ul><li><span>gfan(..., Symmetries => ...), see <span><a href="_gfan.html" title="all initial ideals of an ideal">gfan</a> -- all initial ideals of an ideal</span></span></li>
<li><span>groebnerFan(..., Symmetries => ...), see <span><a href="_groebner__Fan.html" title="the fan of all groebner bases of an ideal">groebnerFan</a> -- the fan of all groebner bases of an ideal</span></span></li>
<li><span>render(..., Symmetries => ...), see <span><a href="_render.html" title="draws the Groebner fan an ideal.">render</a> -- draws the Groebner fan an ideal.</span></span></li>
<li><span>universalGroebnerBasis(..., Symmetries => ...), see <span><a href="_universal__Groebner__Basis.html" title="the union of all reduced Groebner bases of an ideal.">universalGroebnerBasis</a> -- the union of all reduced Groebner bases of an ideal.</span></span></li>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Symmetries.html" title="permutations leaving an ideal invariant to speed up gfan computations.">Symmetries</a> is <span>a <a href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>.</p>
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