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
|
<?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 12: Poincare series</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="chap12" 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="chap11.html">[Previous Chapter]</a> <a href="chap13.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap12_mj.html">[MathJax on]</a></p>
<p><a id="X850CDAFE801E2B2A" name="X850CDAFE801E2B2A"></a></p>
<div class="ChapSects"><a href="chap12.html#X850CDAFE801E2B2A">12 <span class="Heading"> Poincare series</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7CFDEEC07F15CF82">12.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12.html#X84117EA684724D53">12.1-1 EfficientNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12.html#X7EBC620581DCB4D6">12.1-2 ExpansionOfRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12.html#X828B81D9829328F8">12.1-3 PoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12.html#X876B3DFB7B64688C">12.1-4 PoincareSeriesPrimePart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12.html#X7E1A4C8781A02CD0">12.1-5 PoincareSeriesLHS</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12.html#X82CBD11D84D50CBD">12.1-6 Prank</a></span>
</div></div>
</div>
<h3>12 <span class="Heading"> Poincare series</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>12.1 <span class="Heading"> </span></h4>
<p><a id="X84117EA684724D53" name="X84117EA684724D53"></a></p>
<h5>12.1-1 EfficientNormalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EfficientNormalSubgroups</code>( <var class="Arg">G</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">‣ EfficientNormalSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a prime-power group <span class="SimpleMath">G</span> and, optionally, a positive integer <span class="SimpleMath">k</span>. The default is <span class="SimpleMath">k=4</span>. The function returns a list of normal subgroups <span class="SimpleMath">N</span> in <span class="SimpleMath">G</span> such that the Poincare series for <span class="SimpleMath">G</span> equals the Poincare series for the direct product <span class="SimpleMath">(N × (G/N))</span> up to degree <span class="SimpleMath">k</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> </p>
<p><a id="X7EBC620581DCB4D6" name="X7EBC620581DCB4D6"></a></p>
<h5>12.1-2 ExpansionOfRationalFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExpansionOfRationalFunction</code>( <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a positive integer <span class="SimpleMath">n</span> and a rational function <span class="SimpleMath">f(x)=p(x)/q(x)</span> where the degree of the polynomial <span class="SimpleMath">p(x)</span> is less than that of <span class="SimpleMath">q(x)</span>. It returns a list <span class="SimpleMath">[a_0 , a_1 , a_2 , a_3 , ... ,a_n]</span> of the first <span class="SimpleMath">n+1</span> coefficients of the infinite expansion</p>
<p><span class="SimpleMath">f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...</span> .</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> </p>
<p><a id="X828B81D9829328F8" name="X828B81D9829328F8"></a></p>
<h5>12.1-3 PoincareSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var>, <var class="Arg">n</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">‣ PoincareSeries</code>( <var class="Arg">R</var>, <var class="Arg">n</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">‣ PoincareSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</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">‣ PoincareSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. It returns a quotient of polynomials <span class="SimpleMath">f(x)=P(x)/Q(x)</span> whose coefficient of <span class="SimpleMath">x^k</span> equals the rank of the vector space <span class="SimpleMath">H_k(G,Z_p)</span> for all <span class="SimpleMath">k</span> in the range <span class="SimpleMath">k=1</span> to <span class="SimpleMath">k=n</span>. (The second input variable can be omitted, in which case the function tries to choose a "reasonable" value for <span class="SimpleMath">n</span>. For <span class="SimpleMath">2</span>-groups the function PoincareSeriesLHS(G) can be used to produce an <span class="SimpleMath">f(x)</span> that is correct in all degrees.)</p>
<p>In place of the group <span class="SimpleMath">G</span> the function can also input (at least <span class="SimpleMath">n</span> terms of) a minimal mod <span class="SimpleMath">p</span> resolution <span class="SimpleMath">R</span> for <span class="SimpleMath">G</span>.</p>
<p>Alternatively, the first input variable can be a list <span class="SimpleMath">L</span> of integers. In this case the coefficient of <span class="SimpleMath">x^k</span> in <span class="SimpleMath">f(x)</span> is equal to the <span class="SimpleMath">(k+1)</span>st term in the list.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeriesII.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">9</a></span> </p>
<p><a id="X876B3DFB7B64688C" name="X876B3DFB7B64688C"></a></p>
<h5>12.1-4 PoincareSeriesPrimePart</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeriesPrimePart</code>( <var class="Arg">G</var>, <var class="Arg">p</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">G</span>, a prime <span class="SimpleMath">p</span>, and a positive integer <span class="SimpleMath">n</span>. It returns a quotient of polynomials <span class="SimpleMath">f(x)=P(x)/Q(x)</span> whose coefficient of <span class="SimpleMath">x^k</span> equals the rank of the vector space <span class="SimpleMath">H_k(G,Z_p)</span> for all <span class="SimpleMath">k</span> in the range <span class="SimpleMath">k=1</span> to <span class="SimpleMath">k=n</span>.</p>
<p>The efficiency of this function needs to be improved.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">3</a></span> </p>
<p><a id="X7E1A4C8781A02CD0" name="X7E1A4C8781A02CD0"></a></p>
<h5>12.1-5 PoincareSeriesLHS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeriesLHS</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">2</span>-group <span class="SimpleMath">G</span> and returns a quotient of polynomials <span class="SimpleMath">f(x)=P(x)/Q(x)</span> whose coefficient of <span class="SimpleMath">x^k</span> equals the rank of the vector space <span class="SimpleMath">H_k(G,Z_2)</span> for all <span class="SimpleMath">k</span>.</p>
<p>This function was written by <strong class="button">Paul Smith</strong>. It use the Singular system for commutative algebra.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X82CBD11D84D50CBD" name="X82CBD11D84D50CBD"></a></p>
<h5>12.1-6 Prank</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Prank</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and returns the rank of the largest elementary abelian subgroup.</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="chap11.html">[Previous Chapter]</a> <a href="chap13.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>
|
