1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
|
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (HAPprog) - Chapter 1: Resolutions in Hap</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap1" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap0.html">[Previous Chapter]</a> <a href="chap2.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap1_mj.html">[MathJax on]</a></p>
<p><a id="X7C6DD73E7BB931AB" name="X7C6DD73E7BB931AB"></a></p>
<div class="ChapSects"><a href="chap1.html#X7C6DD73E7BB931AB">1 <span class="Heading">Resolutions in Hap</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X804F611B7E23BDB1">1.1 <span class="Heading">The Standard Representation <code class="keyw">HapResolutionRep</code></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8024014D8488FE30">1.2 <span class="Heading">The <code class="keyw">HapLargeGroupResolutionRep</code> Representation</span></a>
</span>
</div>
</div>
<h3>1 <span class="Heading">Resolutions in Hap</span></h3>
<p>This document is only concerned with the representation of resolutions in Hap. Note that it is not a part of Hap. The framework provided here is just an extension of Hap data types used in HAPcryst and HAPprime.</p>
<p>From now on, let <span class="SimpleMath">G</span> be a group and <span class="SimpleMath">dots -> M_n-> M_n-1->dots-> M_1-> M_0-> Z</span> be a resolution with free <span class="SimpleMath">ZG</span> modules <span class="SimpleMath">M_i</span>.</p>
<p>The elements of the modules <span class="SimpleMath">M_i</span> can be represented in different ways. This is what makes different representations for resolutions desirable. First, we will look at the standard representation (<code class="code">HapResolutionRep</code>) as it is defined in Hap. After that, we will present another representation for infinite groups. Note that all non-standard representations must be sub-representations of the standard representation to ensure compatibility with Hap.</p>
<p><a id="X804F611B7E23BDB1" name="X804F611B7E23BDB1"></a></p>
<h4>1.1 <span class="Heading">The Standard Representation <code class="keyw">HapResolutionRep</code></span></h4>
<p>For every <span class="SimpleMath">M_i</span> we fix a basis and number its elements. Furthermore, it is assumed that we have a (partial) enumeration of the group of a resolution. In practice this is done by generating a lookup table on the fly.</p>
<p>In standard representation, the elements of the modules <span class="SimpleMath">M_k</span> are represented by lists -"words"- of pairs of integers. A letter <code class="code">[i,g]</code> of such a word consists of the number of a basis element <code class="code">i</code> or <code class="code">-i</code> for its additive inverse and a number <span class="SimpleMath">g</span> representing a group element.</p>
<p>A <code class="code">HapResolution</code> in <code class="code">HapResolutionRep</code> representation is a component object with the components</p>
<ul>
<li><p><code class="code">group</code>, a group of arbitrary type.</p>
</li>
<li><p><code class="code">elts</code>, a (partial) list of (possibly duplicate) elements in G. This list provides the "enumeration" of the group. Note that there are functions in Hap which assume that <code class="code">elts[1]</code> is the identity element of G.</p>
</li>
<li><p><code class="code">appendToElts(g)</code> a function that appends the group element <code class="code">g</code> to <code class="code">.elts</code>. This is not documented in Hap 1.8.6 but seems to be required for infinite groups. This requirement might vanish in some later version of Hap [G. Ellis, private communication].</p>
</li>
<li><p><code class="code">dimension(k)</code>, a function which returns the ZG-rank of the Module <span class="SimpleMath">M_k</span></p>
</li>
<li><p><code class="code">boundary(k,j)</code>, a function which returns the image in <span class="SimpleMath">M_k-1</span> of the <span class="SimpleMath">j</span>th free generator of <span class="SimpleMath">M_k</span>. Note that negative <span class="SimpleMath">j</span> are valid as input as well. In this case the additive inverse of the boundary of the <span class="SimpleMath">j</span>th generator is returned</p>
</li>
<li><p><code class="code">homotopy(k,[i,g])</code> a function which returns the image in <span class="SimpleMath">M_k+1</span>, under a contracting homotopy <span class="SimpleMath">M_k -> M_k+1</span>, of the element <code class="code">[[i,g]]</code> in <span class="SimpleMath">M_k</span>. The value of this might be <code class="keyw">fail</code>. However, currently (version 1.8.4) some Hap functions assume that <code class="code">homotopy</code> is a function without testing.</p>
</li>
<li><p><code class="code">properties</code>, a list of pairs <code class="code">["name","value"]</code> "name" is a string and value is anything (boolean, number, string...). Every <code class="code">HapResolution</code> (regardless of representation) has to have <code class="code">["type","resolution"]</code>, <code class="code">["length",length]</code> where <code class="code">length</code> is the length of the resolution and <code class="code">["characteristic",char]</code>. Currently (Hap 1.8.6), <code class="code">length</code> must not be <code class="keyw">infinity</code>. The values of these properties can be tested using the Hap function <code class="code">EvaluateProperty(resolution,propertyname)</code>.</p>
</li>
</ul>
<p>Note that making <code class="code">HapResolution</code>s immutable will make the <code class="code">.elts</code> component immutable. As this lookup table might change during calculations, we do not recommend using immutable resolutions (in any representation).</p>
<p><a id="X8024014D8488FE30" name="X8024014D8488FE30"></a></p>
<h4>1.2 <span class="Heading">The <code class="keyw">HapLargeGroupResolutionRep</code> Representation</span></h4>
<p>In this sub-representation of the standard representation, the module elements in this resolution are lists of groupring elements. So the lookup table <code class="code">.elts</code> is not used as long as no conversion to standard representation takes place. In addition to the components of a <code class="keyw">HapResolution</code>, a resolution in large group representation has the following components:</p>
<ul>
<li><p><code class="code">boundary2(resolution,term,gen)</code>, a function that returns the boundary of the <var class="Arg">gen</var>th generator of the <var class="Arg">term</var>th module.</p>
</li>
<li><p><code class="code">groupring</code> the group ring of the resolution <var class="Arg">resolution</var>.</p>
</li>
<li><p><code class="code">dimension2(resolution,term)</code> a function that returns the dimension of the <var class="Arg">term</var>th module of the resolution <var class="Arg">resolution</var>.</p>
</li>
</ul>
<p>The effort of having two versions of <code class="code">boundary</code> and <code class="code">dimension</code> is necessary to keep the structure compatible with the usual Hap resolution.</p>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap0.html">[Previous Chapter]</a> <a href="chap2.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|
