<?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 (repsn) - Chapter 3: Reducible Representations</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="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chapBib.html">Next Chapter</a> </div>
<p><a id="X826CC30186DBDB2B" name="X826CC30186DBDB2B"></a></p>
<div class="ChapSects"><a href="chap3.html#X826CC30186DBDB2B">3 <span class="Heading">Reducible Representations</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7829A125780DD25D">3.1 <span class="Heading">Constituents of Representations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BBA5EC37C52A99D">3.1-1 ConstituentsOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87F71650795FE650">3.1-2 IsReducibleRepresentation</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7FF69B0D7DB36D73">3.2 <span class="Heading">Block Representations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DF3C6607C8EA2BB">3.2-1 EquivalentBlockRepresentation</a></span>
</div>
</div>
<h3>3 <span class="Heading">Reducible Representations</span></h3>
<p>In this chapter we introduce some functions which deal with a complex reducible representation R of a finite group G.</p>
<p><a id="X7829A125780DD25D" name="X7829A125780DD25D"></a></p>
<h4>3.1 <span class="Heading">Constituents of Representations</span></h4>
<p><a id="X7BBA5EC37C52A99D" name="X7BBA5EC37C52A99D"></a></p>
<h5>3.1-1 ConstituentsOfRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ConstituentsOfRepresentation</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>called with a representation <var class="Arg">rep</var> of a group G. This function returns a list of irreducible representations of G which are constituents of <var class="Arg">rep</var>, and their corresponding multiplicities. For example, if <var class="Arg">rep</var> is a representation of G affording a character X such that X = mY + nZ, where Y and Z are irreducible characters of G, and m and n are the corresponding multiplicities, then <code class="code">ConstituentsOfRepresentation</code> returns [[m, S], [n, T]] where S and T are irreducible representations of G affording Y and Z, respectively. This function call can be quite expensive when G is a large group.</p>
<p><a id="X87F71650795FE650" name="X87F71650795FE650"></a></p>
<h5>3.1-2 IsReducibleRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsReducibleRepresentation</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">rep</var> is a representation of a group G then <code class="code">IsReducibleRepresentation</code> returns <code class="code">true</code> if <var class="Arg">rep</var> is a reducible representation of G.</p>
<p><a id="X7FF69B0D7DB36D73" name="X7FF69B0D7DB36D73"></a></p>
<h4>3.2 <span class="Heading">Block Representations</span></h4>
<p><a id="X7DF3C6607C8EA2BB" name="X7DF3C6607C8EA2BB"></a></p>
<h5>3.2-1 EquivalentBlockRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> EquivalentBlockRepresentation</code>( <var class="Arg">rep</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">> EquivalentBlockRepresentation</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">rep</var> is a reducible representation of a group G, this function returns a block diagonal representation of G equivalent to <var class="Arg">rep</var>. If <var class="Arg"> list </var> = [[m1, R1], [m2, R2], ... , [mt, Rt]] is a list of irreducible representations R1, R2, ... , Rt of G with multiplicities m1, m2, ... , mt, then <code class="code">EquivalentBlockRepresentation</code> returns a block diagonal representation of G containing the blocks R1, R2, ... , Rt.</p>
<table class="example">
<tr><td><pre>
gap> G := AlternatingGroup( 5 );;
gap> H := SylowSubgroup( G, 2 );;
gap> chi := TrivialCharacter( H );;
gap> Hrep := IrreducibleAffordingRepresentation( chi );;
gap> rep := InducedSubgroupRepresentation( G, Hrep );;
gap> IsReducibleRepresentation( rep );
true
gap> con := ConstituentsOfRepresentation( rep );
[ [ 1, [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ] ],
[ 1, [ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2 ],
[ 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
[ 1, -E(3), E(3), 0 ],
[ 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2 ] ],
[ [ 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2 ],
[ 0, -E(3), E(3), 1 ],
[ 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
[ 0, 0, 1, 0 ] ] ] ],
[ 2, [ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ -1, 1, 1, 1, -1 ],
[ 0, 0, 0, 0, 1 ],
[ -1, 0, 0, 1, -1 ],
[ 0, 0, 1, 0, 0 ],
[ 0, -1, 0, -1, 1 ] ],
[ [ 0, 0, 0, 0, 1 ],
[ 0, -1, -1, -1, 0 ],
[ 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 0 ],
[ -1, 0, 0, 1, -1 ] ] ] ] ]
gap> EquivalentBlockRepresentation( con );
[ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, -E(3), E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, -1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1 ] ],
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, -E(3), E(3), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ] ] ]
</pre></td></tr></table>
<div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chapBib.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="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>
distrib
>
Mandriva
>
2010.0
>
i586
>
media
>
contrib-release
>
by-pkgid
>
91213ddcfbe7f54821d42c2d9e091326
>
files
>
2562
