Quelle chap3_mj.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/repsn/doc/chap3_mj.html

<?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>
<script type="text/javascript"
  src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<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" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap3"  onload="jscontent()">

<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap2_mj.html">[Previous Chapter]</a>    <a href="chapBib_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap3.html">[MathJax off]</a></p>
<p><a id="X826CC30186DBDB2B" name="X826CC30186DBDB2B"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X826CC30186DBDB2B">3 <span class="Heading">Reducible Representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7829A125780DD25D">3.1 <span class="Heading">Constituents of Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BBA5EC37C52A99D">3.1-1 ConstituentsOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87F71650795FE650">3.1-2 IsReducibleRepresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7FF69B0D7DB36D73">3.2 <span class="Heading">Block Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X847B7C45812A563B">3.2-1 EquivalentBlockRepresentation</a></span>
</div></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 <span class="SimpleMath">\(R\)</span> of a finite group <span class="SimpleMath">\(G\)</span>.</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 <span class="SimpleMath">\(G\)</span>. This function returns a list of irreducible representations of <span class="SimpleMath">\(G\)</span> 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 <span class="SimpleMath">\(G\)</span> affording a character <span class="SimpleMath">\(X\)</span> such that <span class="SimpleMath">\(X = mY + nZ\)</span>, where <span class="SimpleMath">\(Y\)</span> and <span class="SimpleMath">\(Z\)</span> are irreducible characters of <span class="SimpleMath">\(G\)</span>, and <span class="SimpleMath">\(m\)</span> and <span class="SimpleMath">\(n\)</span> are the corresponding multiplicities, then <code class="code">ConstituentsOfRepresentation</code> returns <span class="SimpleMath">\([[m, S]\)</span>, <span class="SimpleMath">\([n, T]]\)</span> where <span class="SimpleMath">\(S\)</span> and <span class="SimpleMath">\(T\)</span> are irreducible representations of <span class="SimpleMath">\(G\)</span> affording <span class="SimpleMath">\(Y\)</span> and <span class="SimpleMath">\(Z\)</span>, respectively. This function call can be quite expensive when <span class="SimpleMath">\(G\)</span> 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 <span class="SimpleMath">\(G\)</span> then <code class="code">IsReducibleRepresentation</code> returns <code class="code">true</code> if <var class="Arg">rep</var> is a reducible representation of <span class="SimpleMath">\(G\)</span>.</p>

<p><a id="X7FF69B0D7DB36D73" name="X7FF69B0D7DB36D73"></a></p>

<h4>3.2 <span class="Heading">Block Representations</span></h4>

<p><a id="X847B7C45812A563B" name="X847B7C45812A563B"></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 <span class="SimpleMath">\(G\)</span>, this function returns a block diagonal representation of <span class="SimpleMath">\(G\)</span> equivalent to <var class="Arg">rep</var>. If <var class="Arg"> list </var> <span class="SimpleMath">\(= [[m1, R1]\)</span>, <span class="SimpleMath">\([m2, R2]\)</span>, ... , <span class="SimpleMath">\([mt, Rt]]\)</span> is a list of irreducible representations <span class="SimpleMath">\(R1\)</span>, <span class="SimpleMath">\(R2\)</span>, ... , <span class="SimpleMath">\(Rt\)</span> of <span class="SimpleMath">\(G\)</span> with multiplicities <span class="SimpleMath">\(m1\)</span>, <span class="SimpleMath">\(m2\)</span>, ... , <span class="SimpleMath">\(mt\)</span>, then <code class="code">EquivalentBlockRepresentation</code> returns a block diagonal representation of <span class="SimpleMath">\(G\)</span> containing the blocks <span class="SimpleMath">\(R1\)</span>, <span class="SimpleMath">\(R2\)</span>, ... , <span class="SimpleMath">\(Rt\)</span>.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := AlternatingGroup( 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SylowSubgroup( G, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">chi := TrivialCharacter( H );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Hrep := IrreducibleAffordingRepresentation( chi );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep := InducedSubgroupRepresentation( G, Hrep );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReducibleRepresentation( rep );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">con := ConstituentsOfRepresentation( rep );</span>
[ [ 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 ] ] ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EquivalentBlockRepresentation( con );</span>
[ (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></div>

<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap2_mj.html">[Previous Chapter]</a>    <a href="chapBib_mj.html">[Next Chapter]</a>   </div>

<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.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>

quality100%

¤ Dauer der Verarbeitung: 0.11 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.