<?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>flip(Module,Module) -- matrix of commutativity of tensor product</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_floor.html">next</a> | <a href="_flatten__Ring.html">previous</a> | <a href="_floor.html">forward</a> | <a href="_flatten__Ring.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>flip(Module,Module) -- matrix of commutativity of tensor product</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>flip(F,G)</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_flip_lp__Module_cm__Module_rp.html" title="matrix of commutativity of tensor product">flip</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
<li><span><tt>G</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the matrix representing the natural isomorphism <tt>G ** F <-- F ** G</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
</td></tr>
<tr><td><pre>i2 : F = R^{1,2,3}
3
o2 = R
o2 : R-module, free, degrees {-1, -2, -3}</pre>
</td></tr>
<tr><td><pre>i3 : G = R^{10,20,30}
3
o3 = R
o3 : R-module, free, degrees {-10, -20, -30}</pre>
</td></tr>
<tr><td><pre>i4 : f = flip(F,G)
o4 = {-11} | 1 0 0 0 0 0 0 0 0 |
{-12} | 0 0 0 1 0 0 0 0 0 |
{-13} | 0 0 0 0 0 0 1 0 0 |
{-21} | 0 1 0 0 0 0 0 0 0 |
{-22} | 0 0 0 0 1 0 0 0 0 |
{-23} | 0 0 0 0 0 0 0 1 0 |
{-31} | 0 0 1 0 0 0 0 0 0 |
{-32} | 0 0 0 0 0 1 0 0 0 |
{-33} | 0 0 0 0 0 0 0 0 1 |
9 9
o4 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i5 : isHomogeneous f
o5 = true</pre>
</td></tr>
<tr><td><pre>i6 : target f
9
o6 = R
o6 : R-module, free, degrees {-11, -12, -13, -21, -22, -23, -31, -32, -33}</pre>
</td></tr>
<tr><td><pre>i7 : source f
9
o7 = R
o7 : R-module, free, degrees {-11, -21, -31, -12, -22, -32, -13, -23, -33}</pre>
</td></tr>
<tr><td><pre>i8 : target f === G**F
o8 = true</pre>
</td></tr>
<tr><td><pre>i9 : source f === F**G
o9 = true</pre>
</td></tr>
<tr><td><pre>i10 : u = x * F_0
o10 = {-1} | x |
{-2} | 0 |
{-3} | 0 |
3
o10 : R</pre>
</td></tr>
<tr><td><pre>i11 : v = y * G_1
o11 = {-10} | 0 |
{-20} | y |
{-30} | 0 |
3
o11 : R</pre>
</td></tr>
<tr><td><pre>i12 : u ** v
o12 = {-11} | 0 |
{-21} | xy |
{-31} | 0 |
{-12} | 0 |
{-22} | 0 |
{-32} | 0 |
{-13} | 0 |
{-23} | 0 |
{-33} | 0 |
9
o12 : R</pre>
</td></tr>
<tr><td><pre>i13 : v ** u
o13 = {-11} | 0 |
{-12} | 0 |
{-13} | 0 |
{-21} | xy |
{-22} | 0 |
{-23} | 0 |
{-31} | 0 |
{-32} | 0 |
{-33} | 0 |
9
o13 : R</pre>
</td></tr>
<tr><td><pre>i14 : f * (u ** v)
o14 = {-11} | 0 |
{-12} | 0 |
{-13} | 0 |
{-21} | xy |
{-22} | 0 |
{-23} | 0 |
{-31} | 0 |
{-32} | 0 |
{-33} | 0 |
9
o14 : R</pre>
</td></tr>
<tr><td><pre>i15 : f * (u ** v) === v ** u
o15 = true</pre>
</td></tr>
</table>
</div>
</div>
</div>
</body>
</html>
