<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>dualize -- The dual of a face or complex.</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_edim.html">next</a> | <a href="_dual__Grading.html">previous</a> | <a href="_edim.html">forward</a> | <a href="_dual__Grading.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>dualize -- The dual of a face or complex.</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>dualize(F)</tt><br/><tt>dualize(C)</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="___Face.html">face</a></span></span></li>
<li><span><tt>C</tt>, <span>an <a href="___Complex.html">embedded complex</a></span></span></li>
<li><span><tt>C</tt>, <span>an <a href="___Co__Complex.html">embedded co-complex</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Co__Complex.html">embedded co-complex</a></span></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>Returns the dual of a face or a (co)complex. This is in the sense of dual face of Polytopes, so the faces of C have to be faces of a polytope.</p>
<p>The dual (co)complex dC is stored in C.dualComplex=dC and dC.dualComplex=C.</p>
<p>Note that if C is a Stanley-Reisner subcomplex of a simplex then dualize complement C is the isomorphic geometric complex of strata.</p>
<div/>
<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : addCokerGrading R
o2 = | -1 -1 -1 -1 |
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
5 4
o2 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i3 : C=simplex R
o3 = 4: x x x x x
0 1 2 3 4
o3 : complex of dim 4 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0</pre>
</td></tr>
<tr><td><pre>i4 : bC=boundaryOfPolytope C
o4 = 3: x x x x x x x x x x x x x x x x x x x x
0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 1 2 3 4
o4 : complex of dim 3 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1</pre>
</td></tr>
<tr><td><pre>i5 : F=bC.fc_2_0
o5 = x x x
0 1 2
o5 : face with 3 vertices</pre>
</td></tr>
<tr><td><pre>i6 : coordinates F
o6 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}}
o6 : List</pre>
</td></tr>
<tr><td><pre>i7 : dualize F
o7 = v v
0 1
o7 : face with 2 vertices</pre>
</td></tr>
<tr><td><pre>i8 : coordinates dualize F
o8 = {{-1, -1, -1, 4}, {-1, -1, 4, -1}}
o8 : List</pre>
</td></tr>
<tr><td><pre>i9 : dbC=dualize bC
o9 = 0: v v v v v
0 1 2 3 4
o9 : co-complex of dim 0 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {0, 5, 10, 10, 5, 1}, Euler = 1</pre>
</td></tr>
<tr><td><pre>i10 : complement F
o10 = x x
3 4
o10 : face with 2 vertices</pre>
</td></tr>
<tr><td><pre>i11 : coordinates complement F
o11 = {{0, 0, 1, 0}, {0, 0, 0, 1}}
o11 : List</pre>
</td></tr>
<tr><td><pre>i12 : complement bC
o12 = 0: x x x x x
4 3 2 1 0
o12 : co-complex of dim 0 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {0, 5, 10, 10, 5, 1}, Euler = 1</pre>
</td></tr>
<tr><td><pre>i13 : dualize complement bC
o13 = 3: v v v v v v v v v v v v v v v v v v v v
1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 4 0 1 2 3
o13 : complex of dim 3 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1</pre>
</td></tr>
<tr><td><pre>i14 : bC
o14 = 3: x x x x x x x x x x x x x x x x x x x x
0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 1 2 3 4
o14 : complex of dim 3 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1</pre>
</td></tr>
<tr><td><pre>i15 : coordinates dualize complement F
o15 = {{-1, 4, -1, -1}, {4, -1, -1, -1}, {-1, -1, -1, -1}}
o15 : List</pre>
</td></tr>
<tr><td><pre>i16 : coordinates F
o16 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}}
o16 : List</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_complement_lp__Face_rp.html" title="Compute the complement face of a simplex.">complement(Face)</a> -- Compute the complement face of a simplex.</span></li>
<li><span><a href="_complement_lp__Complex_rp.html" title="Compute the complement CoComplex.">complement(Complex)</a> -- Compute the complement CoComplex.</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>dualize</tt> :</h2>
<ul><li>dualize(Complex)</li>
<li>dualize(Face)</li>
</ul>
</div>
</div>
</body>
</html>
