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
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
|
<?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 (HAP commands) - Chapter 18: Orbit polytopes and fundamental domains</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="chap18" 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> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap17.html">[Previous Chapter]</a> <a href="chap19.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap18_mj.html">[MathJax on]</a></p>
<p><a id="X7CD67FEA7A1B6345" name="X7CD67FEA7A1B6345"></a></p>
<div class="ChapSects"><a href="chap18.html#X7CD67FEA7A1B6345">18 <span class="Heading"> Orbit polytopes and fundamental domains</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X7CFDEEC07F15CF82">18.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X802CD86983E8384B">18.1-1 CoxeterComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7B60301179A6E7D2">18.1-2 ContractibleGcomplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X82B7AF6A876AA021">18.1-3 QuotientOfContractibleGcomplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X8092E70D8425CEBB">18.1-4 TruncatedGComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X79F4F7938116201E">18.1-5 FundamentalDomainStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X80EC50C27EFF2E12">18.1-6 OrbitPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7A4234867B232E34">18.1-7 PolytopalComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7E03550580383C01">18.1-8 PolytopalGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X797A90E17C18FC89">18.1-9 VectorStabilizer</a></span>
</div></div>
</div>
<h3>18 <span class="Heading"> Orbit polytopes and fundamental domains</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>18.1 <span class="Heading"> </span></h4>
<p><a id="X802CD86983E8384B" name="X802CD86983E8384B"></a></p>
<h5>18.1-1 CoxeterComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterComplex</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterComplex</code>( <var class="Arg">D</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> of finite type. It returns a non-free ZW-resolution for the associated Coxeter group <span class="SimpleMath">W</span>. The non-free resolution is obtained from the permutahedron of type <span class="SimpleMath">W</span>. A positive integer <span class="SimpleMath">n</span> can be entered as an optional second variable; just the first <span class="SimpleMath">n</span> terms of the non-free resolution are then returned.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> </p>
<p><a id="X7B60301179A6E7D2" name="X7B60301179A6E7D2"></a></p>
<h5>18.1-2 ContractibleGcomplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContractibleGcomplex</code>( <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs one of the following strings <span class="SimpleMath">str</span>=: <br /> <br /> "SL(2,Z)" , "SL(3,Z)" , "PGL(3,Z[i])" , "PGL(3,Eisenstein_Integers)" , "PSL(4,Z)" , "PSL(4,Z)_b" , "PSL(4,Z)_c" , "PSL(4,Z)_d" , "Sp(4,Z)" <br /> <br /> or one of the following strings <br /> <br /> "SL(2,O-2)" , "SL(2,O-7)" , "SL(2,O-11)" , "SL(2,O-19)" , "SL(2,O-43)" , "SL(2,O-67)" , "SL(2,O-163)" <br /> <br /> It returns a non-free ZG-resolution for the group <span class="SimpleMath">G</span> described by the string. The stabilizer groups of cells are finite. (Subscripts _b , _c , _d denote alternative non-free ZG-resolutions for a given group G.)<br /> <br /> Data for the first list of non-free resolutions was provided provided by <strong class="button">Mathieu Dutour</strong>. Data for the second list was provided by <strong class="button">Alexander Rahm</strong>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap6.html">2</a></span> , <span class="URL"><a href="../tutorial/chap7.html">3</a></span> , <span class="URL"><a href="../tutorial/chap9.html">4</a></span> , <span class="URL"><a href="../tutorial/chap11.html">5</a></span> , <span class="URL"><a href="../tutorial/chap13.html">6</a></span> , <span class="URL"><a href="../tutorial/chap14.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutBredon.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">10</a></span> </p>
<p><a id="X82B7AF6A876AA021" name="X82B7AF6A876AA021"></a></p>
<h5>18.1-3 QuotientOfContractibleGcomplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuotientOfContractibleGcomplex</code>( <var class="Arg">C</var>, <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">C</span> and a finite subgroup <span class="SimpleMath">D</span> of <span class="SimpleMath">G</span> which is a subgroup of each cell stabilizer group for <span class="SimpleMath">C</span>. Each element of <span class="SimpleMath">D</span> must preserves the orientation of any cell stabilized by it. It returns the corresponding non-free <span class="SimpleMath">Z(G/D)</span>-resolution. (So, for instance, from the <span class="SimpleMath">SL(2,O)</span> complex <span class="SimpleMath">C=ContractibleGcomplex("SL(2,O-2)");</span> we can construct a <span class="SimpleMath">PSL(2,O)</span>-complex using this function.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap13.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">2</a></span> </p>
<p><a id="X8092E70D8425CEBB" name="X8092E70D8425CEBB"></a></p>
<h5>18.1-4 TruncatedGComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TruncatedGComplex</code>( <var class="Arg">R</var>, <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> and two positive integers <span class="SimpleMath">m</span> and <span class="SimpleMath">n</span>. It returns the non-free <span class="SimpleMath">ZG</span>-resolution consisting of those modules in <span class="SimpleMath">R</span> of degree at least <span class="SimpleMath">m</span> and at most <span class="SimpleMath">n</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X79F4F7938116201E" name="X79F4F7938116201E"></a></p>
<h5>18.1-5 FundamentalDomainStandardSpaceGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FundamentalDomainStandardSpaceGroup</code>( <var class="Arg">v</var>, <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a crystallographic group G (represented using AffineCrystGroupOnRight as in the GAP package Cryst). It also inputs a choice of vector v in the euclidean space <span class="SimpleMath">R^n</span> on which <span class="SimpleMath">G</span> acts. It returns the Dirichlet-Voronoi fundamental cell for the action of <span class="SimpleMath">G</span> on euclidean space corresponding to the vector <span class="SimpleMath">v</span>. The fundamental cell is a fundamental domain if <span class="SimpleMath">G</span> is Bieberbach. The fundamental cell/domain is returned as a <q>Polymake object</q>. Currently the function only applies to certain crystallographic groups. See the manuals to HAPcryst and HAPpolymake for full details.</p>
<p>This function is part of the HAPcryst package written by <strong class="button">Marc Roeder</strong> and is thus only available if HAPcryst is loaded.</p>
<p>The function requires the use of Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">2</a></span> </p>
<p><a id="X80EC50C27EFF2E12" name="X80EC50C27EFF2E12"></a></p>
<h5>18.1-6 OrbitPolytope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitPolytope</code>( <var class="Arg">G</var>, <var class="Arg">v</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector <span class="SimpleMath">v</span> of length <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span>. Let <span class="SimpleMath">P(G,v)</span> be the convex polytope arising as the convex hull of the Euclidean points in the orbit of <span class="SimpleMath">v</span> under the action of <span class="SimpleMath">G</span>. The function also inputs a sublist <span class="SimpleMath">L</span> of the following list of strings:</p>
<p>["dimension","vertex_degree", "visual_graph", "schlegel","visual"]</p>
<p>Depending on the sublist, the function:</p>
<ul>
<li><p>prints the dimension of the orbit polytope <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>prints the degree of a vertex in the graph of <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>visualizes the graph of <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>visualizes the Schlegel diagram of <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>visualizes <span class="SimpleMath">P(G,v)</span> if the polytope is of dimension 2 or 3.</p>
</li>
</ul>
<p>The function uses Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">2</a></span> </p>
<p><a id="X7A4234867B232E34" name="X7A4234867B232E34"></a></p>
<h5>18.1-7 PolytopalComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolytopalComplex</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolytopalComplex</code>( <var class="Arg">G</var>, <var class="Arg">v</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector <span class="SimpleMath">v</span> of length <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span>. Let <span class="SimpleMath">P(G,v)</span> be the convex polytope arising as the convex hull of the Euclidean points in the orbit of <span class="SimpleMath">v</span> under the action of <span class="SimpleMath">G</span>. The cellular chain complex <span class="SimpleMath">C_*=C_*(P(G,v))</span> is an exact sequence of (not necessarily free) <span class="SimpleMath">ZG</span>-modules. The function returns a component object <span class="SimpleMath">R</span> with components:</p>
<ul>
<li><p><span class="SimpleMath">R!.dimension(k)</span> is a function which returns the number of <span class="SimpleMath">G</span>-orbits of the <span class="SimpleMath">k</span>-dimensional faces in <span class="SimpleMath">P(G,v)</span>. If each <span class="SimpleMath">k</span>-face has trivial stabilizer subgroup in <span class="SimpleMath">G</span> then <span class="SimpleMath">C_k</span> is a free <span class="SimpleMath">ZG</span>-module of rank <span class="SimpleMath">R.dimension(k)</span>.</p>
</li>
<li><p><span class="SimpleMath">R!.stabilizer(k,n)</span> is a function which returns the stabilizer subgroup for a face in the <span class="SimpleMath">n</span>-th orbit of <span class="SimpleMath">k</span>-faces.</p>
</li>
<li><p>If all faces of dimension <<span class="SimpleMath">k+1</span> have trivial stabilizer group then the first <span class="SimpleMath">k</span> terms of <span class="SimpleMath">C_*</span> constitute part of a free <span class="SimpleMath">ZG</span>-resolution. The boundary map is described by the function <span class="SimpleMath">boundary(k,n)</span> . (If some faces have non-trivial stabilizer group then <span class="SimpleMath">C_*</span> is not free and no attempt is made to determine signs for the boundary map.)</p>
</li>
<li><p><span class="SimpleMath">R!.elements</span>, <span class="SimpleMath">R!.group</span>, <span class="SimpleMath">R!.properties</span> are as in a <span class="SimpleMath">ZG</span>-resolution.</p>
</li>
</ul>
<p>If an optional third input variable <span class="SimpleMath">n</span> is used, then only the first <span class="SimpleMath">n</span> terms of the resolution <span class="SimpleMath">C_*</span> will be computed.</p>
<p>The function uses Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">2</a></span> </p>
<p><a id="X7E03550580383C01" name="X7E03550580383C01"></a></p>
<h5>18.1-8 PolytopalGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolytopalGenerators</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector <span class="SimpleMath">v</span> of length <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span>, and the vector <span class="SimpleMath">v</span> must be chosen so that it has trivial stabilizer subgroup in <span class="SimpleMath">G</span>. Let <span class="SimpleMath">P(G,v)</span> be the convex polytope arising as the convex hull of the Euclidean points in the orbit of <span class="SimpleMath">v</span> under the action of <span class="SimpleMath">G</span>. The function returns a record <span class="SimpleMath">P</span> with components:</p>
<ul>
<li><p><span class="SimpleMath">P.generators</span> is a list of all those elements <span class="SimpleMath">g</span> in <span class="SimpleMath">G</span> such that <span class="SimpleMath">g⋅ v</span> has an edge in common with <span class="SimpleMath">v</span>. The list is a generating set for <span class="SimpleMath">G</span>.</p>
</li>
<li><p><span class="SimpleMath">P.vector</span> is the vector <span class="SimpleMath">v</span>.</p>
</li>
<li><p><span class="SimpleMath">P.hasseDiagram</span> is the Hasse diagram of the cone at <span class="SimpleMath">v</span>.</p>
</li>
</ul>
<p>The function uses Polymake software. The function is joint work with Seamus Kelly.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X797A90E17C18FC89" name="X797A90E17C18FC89"></a></p>
<h5>18.1-9 VectorStabilizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorStabilizer</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector of degree <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span> and the function returns the group of elements in <span class="SimpleMath">G</span> that fix <span class="SimpleMath">v</span>.</p>
<p><strong class="button">Examples:</strong></p>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap17.html">[Previous Chapter]</a> <a href="chap19.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> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|
