Sophie: gap-system-packages-4.4.12-5mdv2010.0 i586

gap-system-packages-4.4.12-5mdv2010.0.i586.rpm

<?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 (Wedderga) - Chapter 4: Idempotents</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" />
</head>
<body>


<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="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop">&nbsp;<a href="chap0.html">Top of Book</a>&nbsp;  &nbsp;<a href="chap3.html">Previous Chapter</a>&nbsp;  &nbsp;<a href="chap5.html">Next Chapter</a>&nbsp;  </div>

<p><a id="X7C651C9C78398FFF" name="X7C651C9C78398FFF"></a></p>
<div class="ChapSects"><a href="chap4.html#X7C651C9C78398FFF">4 <span class="Heading">Idempotents</span></a>
<div class="ContSect"><span class="nocss">&nbsp;</span><a href="chap4.html#X7DF49142844C278D">4.1 <span class="Heading">Computing idempotents from character table</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X7BBEB4A084DBF0D6">4.1-1 PrimitiveCentralIdempotentsByCharacterTable</a></span>
</div>
<div class="ContSect"><span class="nocss">&nbsp;</span><a href="chap4.html#X83F7CF1E87D02581">4.2 <span class="Heading">Testing lists of idempotents for completeness</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X81FCD27E812078F0">4.2-1 IsCompleteSetOfOrthogonalIdempotents</a></span>
</div>
<div class="ContSect"><span class="nocss">&nbsp;</span><a href="chap4.html#X7C66102485AF5F80">4.3 <span class="Heading">Idempotents from Shoda pairs</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X7B48EE1A7ECAB151">4.3-1 PrimitiveCentralIdempotentsByStrongSP</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X82460B1285A0A7D7">4.3-2 PrimitiveCentralIdempotentsBySP</a></span>
</div>
</div>

<h3>4 <span class="Heading">Idempotents</span></h3>

<p><a id="X7DF49142844C278D" name="X7DF49142844C278D"></a></p>

<h4>4.1 <span class="Heading">Computing idempotents from character table</span></h4>

<p><a id="X7BBEB4A084DBF0D6" name="X7BBEB4A084DBF0D6"></a></p>

<h5>4.1-1 PrimitiveCentralIdempotentsByCharacterTable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&gt; PrimitiveCentralIdempotentsByCharacterTable</code>( <var class="Arg">FG</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>A list of group algebra elements.</p>

<p>The input <var class="Arg">FG</var> should be a semisimple group algebra.</p>

<p>Returns the list of primitive central idempotents of <var class="Arg">FG</var> using the character table of G (<a href="chap7.html#X87B6505C7C2EE054"><b>7.4</b></a>).</p>


<table class="example">
<tr><td><pre>

gap&gt; QS3 := GroupRing( Rationals, SymmetricGroup(3) );;                 
gap&gt; PrimitiveCentralIdempotentsByCharacterTable( QS3 );
[ (1/6)*()+(-1/6)*(2,3)+(-1/6)*(1,2)+(1/6)*(1,2,3)+(1/6)*(1,3,2)+(-1/6)*(1,3),
  (2/3)*()+(-1/3)*(1,2,3)+(-1/3)*(1,3,2), (1/6)*()+(1/6)*(2,3)+(1/6)*(1,2)+(1/
    6)*(1,2,3)+(1/6)*(1,3,2)+(1/6)*(1,3) ]
gap&gt; QG:=GroupRing( Rationals , SmallGroup(24,3) );
&lt;algebra-with-one over Rationals, with 4 generators&gt;
gap&gt; FG:=GroupRing( CF(3) , SmallGroup(24,3) );
&lt;algebra-with-one over CF(3), with 4 generators&gt;
gap&gt; pciQG := PrimitiveCentralIdempotentsByCharacterTable(QG);;
gap&gt; pciFG := PrimitiveCentralIdempotentsByCharacterTable(FG);;
gap&gt; Length(pciQG);
5
gap&gt; Length(pciFG);
7

</pre></td></tr></table>

<p><a id="X83F7CF1E87D02581" name="X83F7CF1E87D02581"></a></p>

<h4>4.2 <span class="Heading">Testing lists of idempotents for completeness</span></h4>

<p><a id="X81FCD27E812078F0" name="X81FCD27E812078F0"></a></p>

<h5>4.2-1 IsCompleteSetOfOrthogonalIdempotents</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&gt; IsCompleteSetOfOrthogonalIdempotents</code>( <var class="Arg">R, list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The input should be formed by a unital ring <var class="Arg">R</var> and a list <var class="Arg">list</var> of elements of <var class="Arg">R</var>.</p>

<p>Returns <code class="keyw">true</code> if the list <var class="Arg">list</var> is a complete list of orthogonal idempotents of <var class="Arg">R</var>. That is, the output is <code class="keyw">true</code> provided the following conditions are satisfied:</p>

<p>* The sum of the elements of <var class="Arg">list</var> is the identity of <var class="Arg">R</var>,</p>

<p>* e^2=e, for every e in <var class="Arg">list</var> and</p>

<p>* e*f=0, if e and f are elements in different positions of <var class="Arg">list</var>.</p>

<p>No claim is made on the idempotents being central or primitive.</p>

<p>Note that the if a non-zero element t of <var class="Arg">R</var> appears in two different positions of <var class="Arg">list</var> then the output is <code class="keyw">false</code>, and that the list <var class="Arg">list</var> must not contain zeroes.</p>


<table class="example">
<tr><td><pre>

gap&gt; QS5 := GroupRing( Rationals, SymmetricGroup(5) );;
gap&gt; idemp := PrimitiveCentralIdempotentsByCharacterTable( QS5 );;
gap&gt; IsCompleteSetOfOrthogonalIdempotents( QS5, idemp );
true
gap&gt; IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ) ] );
true
gap&gt; IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ), One( QS5 ) ] );
false

</pre></td></tr></table>

<p><a id="X7C66102485AF5F80" name="X7C66102485AF5F80"></a></p>

<h4>4.3 <span class="Heading">Idempotents from Shoda pairs</span></h4>

<p><a id="X7B48EE1A7ECAB151" name="X7B48EE1A7ECAB151"></a></p>

<h5>4.3-1 PrimitiveCentralIdempotentsByStrongSP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&gt; PrimitiveCentralIdempotentsByStrongSP</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>A list of group algebra elements.</p>

<p>The input <var class="Arg">FG</var> should be a semisimple group algebra of a finite group G whose coefficient field F is either a finite field or the field ℚ of rationals.</p>

<p>If F = ℚ then the output is the list of primitive central idempotents of the group algebra <var class="Arg">FG</var> realizable by strong Shoda pairs (<a href="chap7.html#X81DAF5267D30C83A"><b>7.15</b></a>) of G.</p>

<p>If F is a finite field then the output is the list of primitive central idempotents of <var class="Arg">FG</var> realizable by strong Shoda pairs (K,H) of G and q-cyclotomic classes modulo the index of H in K (<a href="chap7.html#X800D8C5087D79DC8"><b>7.17</b></a>).</p>

<p>If the list of primitive central idempotents given by the output is not complete (i.e. if the group G is not <em>strongly monomial</em> (<a href="chap7.html#X84C694978557EFE5"><b>7.16</b></a>)) then a warning is displayed.</p>


<table class="example">
<tr><td><pre>

gap&gt; QG:=GroupRing( Rationals, AlternatingGroup(4) );;           
gap&gt; PrimitiveCentralIdempotentsByStrongSP( QG );
[ (1/12)*()+(1/12)*(2,3,4)+(1/12)*(2,4,3)+(1/12)*(1,2)(3,4)+(1/12)*(1,2,3)+(1/
    12)*(1,2,4)+(1/12)*(1,3,2)+(1/12)*(1,3,4)+(1/12)*(1,3)(2,4)+(1/12)*
    (1,4,2)+(1/12)*(1,4,3)+(1/12)*(1,4)(2,3),
  (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(-1/12)*(1,2,3)+(
    -1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)(2,4)+(-1/12)*
    (1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3),
  (3/4)*()+(-1/4)*(1,2)(3,4)+(-1/4)*(1,3)(2,4)+(-1/4)*(1,4)(2,3) ]
gap&gt; QG := GroupRing( Rationals, SmallGroup(24,3) );;
gap&gt; PrimitiveCentralIdempotentsByStrongSP( QG );;
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input! 
gap&gt; FG := GroupRing( GF(2), Group((1,2,3)) );;
gap&gt; PrimitiveCentralIdempotentsByStrongSP( FG );
[ (Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2), 
  (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ]
gap&gt; FG := GroupRing( GF(5), SmallGroup(24,3) );; 
gap&gt; PrimitiveCentralIdempotentsByStrongSP( FG );;
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input! 

</pre></td></tr></table>

<p><a id="X82460B1285A0A7D7" name="X82460B1285A0A7D7"></a></p>

<h5>4.3-2 PrimitiveCentralIdempotentsBySP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&gt; PrimitiveCentralIdempotentsBySP</code>( <var class="Arg">QG</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>A list of group algebra elements.</p>

<p>The input should be a rational group algebra of a finite group G.</p>

<p>Returns a list containing all the primitive central idempotents e of the rational group algebra <var class="Arg">QG</var> such that chi(e)ne 0 for some irreducible monomial character chi of G.</p>

<p>The output is the list of all primitive central idempotents of <var class="Arg">QG</var> if and only if G is monomial, otherwise a warning message is displayed.</p>


<table class="example">
<tr><td><pre>

gap&gt; QG := GroupRing( Rationals, SymmetricGroup(4) );
&lt;algebra-with-one over Rationals, with 2 generators&gt;
gap&gt; pci:=PrimitiveCentralIdempotentsBySP( QG );
[ (1/24)*()+(1/24)*(3,4)+(1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*(2,4,3)+(1/24)*
    (2,4)+(1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(1/24)*(1,2,3,4)+(1/
    24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(1/24)*(1,3,4,2)+(1/24)*
    (1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(1/24)*(1,3,2,4)+(1/24)*(1,4,3,2)+(
    1/24)*(1,4,2)+(1/24)*(1,4,3)+(1/24)*(1,4)+(1/24)*(1,4,2,3)+(1/24)*(1,4)
    (2,3), (1/24)*()+(-1/24)*(3,4)+(-1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*
    (2,4,3)+(-1/24)*(2,4)+(-1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(-1/
    24)*(1,2,3,4)+(-1/24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(-1/24)*
    (1,3,4,2)+(-1/24)*(1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(-1/24)*
    (1,3,2,4)+(-1/24)*(1,4,3,2)+(1/24)*(1,4,2)+(1/24)*(1,4,3)+(-1/24)*(1,4)+(
    -1/24)*(1,4,2,3)+(1/24)*(1,4)(2,3), (3/8)*()+(-1/8)*(3,4)+(-1/8)*(2,3)+(
    -1/8)*(2,4)+(-1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(1/8)*(1,2,3,4)+(1/8)*
    (1,2,4,3)+(1/8)*(1,3,4,2)+(-1/8)*(1,3)+(-1/8)*(1,3)(2,4)+(1/8)*(1,3,2,4)+(
    1/8)*(1,4,3,2)+(-1/8)*(1,4)+(1/8)*(1,4,2,3)+(-1/8)*(1,4)(2,3), 
  (3/8)*()+(1/8)*(3,4)+(1/8)*(2,3)+(1/8)*(2,4)+(1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(
    -1/8)*(1,2,3,4)+(-1/8)*(1,2,4,3)+(-1/8)*(1,3,4,2)+(1/8)*(1,3)+(-1/8)*(1,3)
    (2,4)+(-1/8)*(1,3,2,4)+(-1/8)*(1,4,3,2)+(1/8)*(1,4)+(-1/8)*(1,4,2,3)+(-1/
    8)*(1,4)(2,3), (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(
    -1/12)*(1,2,3)+(-1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)
    (2,4)+(-1/12)*(1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3) ]
gap&gt; IsCompleteSetOfPCIs(QG,pci);
true
gap&gt; QS5 := GroupRing( Rationals, SymmetricGroup(5) );;
gap&gt; pci:=PrimitiveCentralIdempotentsBySP( QS5 );;
Wedderga: Warning!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
gap&gt; IsCompleteSetOfPCIs( QS5 , pci );
false

</pre></td></tr></table>

<p>The output of <code class="func">PrimitiveCentralIdempotentsBySP</code> contains the output of <code class="func">PrimitiveCentralIdempotentsByStrongSP</code> (<a href="chap4.html#X7B48EE1A7ECAB151"><b>4.3-1</b></a>), possibly properly.</p>


<table class="example">
<tr><td><pre>

gap&gt; QG := GroupRing( Rationals, SmallGroup(48,28) );;
gap&gt; pci:=PrimitiveCentralIdempotentsBySP( QG );;
Wedderga: Warning!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input! 
gap&gt; Length(pci);    
6
gap&gt; spci:=PrimitiveCentralIdempotentsByStrongSP( QG );;  
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input! 
gap&gt; Length(spci);
5
gap&gt; IsSubset(pci,spci);          
true
gap&gt; QG:=GroupRing(Rationals,SmallGroup(1000,86));
&lt;algebra-with-one over Rationals, with 6 generators&gt;
gap&gt; IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsBySP(QG) );
true
gap&gt; IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsByStrongSP(QG) );
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
false

</pre></td></tr></table>


<div class="chlinkprevnextbot">&nbsp;<a href="chap0.html">Top of Book</a>&nbsp;  &nbsp;<a href="chap3.html">Previous Chapter</a>&nbsp;  &nbsp;<a href="chap5.html">Next Chapter</a>&nbsp;  </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="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>