Gedichte Musik Bilder Quellcodebibliothek Diashow Normaldarstellung
Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/majoranaalgebras/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 7.6.2024 mit Größe 8 kB image not shown  

Quelle  chap5.html   Sprache: HTML

 
 products/Sources/formale Sprachen/GAP/pkg/majoranaalgebras/doc/chap5.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>
<title>GAP (MajoranaAlgebras) - Chapter 5: Functions for testing Majorana 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="chap5"  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="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="chap4.html">[Previous Chapter]</a>    <a href="chap6.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap5_mj.html">[MathJax on]</a></p>
<p><a id="X7CB84E157B2F902D" name="X7CB84E157B2F902D"></a></p>
<div class="ChapSects"><a href="chap5.html#X7CB84E157B2F902D">5 <span class="Heading">Functions for testing Majorana representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80ABA2918548E108">5.1 <span class="Heading">The main function</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X867D4BF17C730633">5.1-1 MajoranaAlgebraTest</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X87C3D1B984960984">5.2 <span class="Heading">Other functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7AD4DBC27DB539EF">5.2-1 MAJORANA_TestFrobeniusForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E3499067808B4A0">5.2-2 MAJORANA_TestInnerProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CB2F52386F07B68">5.2-3 MAJORANA_TestAxiomM2</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X85DFEF167A236627">5.2-4 MAJORANA_TestPrimitivity</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Functions for testing Majorana representations</span></h3>

<p>The output of the function <code class="func">MajoranaRepresentation</code> (<a href="chap3.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>) is guaranteed to be a commutative algebra generated by idempotents whose eigenspaces obey the Majorana fusion law. To check that the output is truly a Majorana algebra, one must also check that</p>


<ul>
<li><p>the inner product is a Frobenius form (see <code class="func">MAJORANA_TestFrobeniusForm</code> (<a href="chap5.html#X7AD4DBC27DB539EF"><span class="RefLink">5.2-1</span></a>));</p>

</li>
<li><p>the inner product is positive definite (see <code class="func">MAJORANA_TestInnerProduct</code> (<a href="chap5.html#X7E3499067808B4A0"><span class="RefLink">5.2-2</span></a>));</p>

</li>
<li><p>the inner product obeys axiom M2 (Norton's inequality) (see MAJORANA_TestAxiomM2 (5.2-3));



</li>
<li><p>the algebra is primitive (see <code class="func">MAJORANA_TestPrimitivity</code> (<a href="chap5.html#X85DFEF167A236627"><span class="RefLink">5.2-4</span></a>)).</p>

</li>
</ul>
<p><a id="X80ABA2918548E108" name="X80ABA2918548E108"></a></p>

<h4>5.1 <span class="Heading">The main function</span></h4>

<p><a id="X867D4BF17C730633" name="X867D4BF17C730633"></a></p>

<h5>5.1-1 MajoranaAlgebraTest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MajoranaAlgebraTest</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the algebra given by <var class="Arg">rep</var> is indeed a Majorana algebra.</p>

<p>Note: does not check that the algebra obeys axiom M2 (Norton's inequality), this can be separately tested using MAJORANA_TestAxiomM2 (5.2-3).



<p><a id="X87C3D1B984960984" name="X87C3D1B984960984"></a></p>

<h4>5.2 <span class="Heading">Other functions</span></h4>

<p><a id="X7AD4DBC27DB539EF" name="X7AD4DBC27DB539EF"></a></p>

<h5>5.2-1 MAJORANA_TestFrobeniusForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestFrobeniusForm</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the inner product given by <var class="Arg">rep.innerproducts</var> is a Frobenius form, otherwise returns false.</p>

<p><a id="X7E3499067808B4A0" name="X7E3499067808B4A0"></a></p>

<h5>5.2-2 MAJORANA_TestInnerProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestInnerProduct</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the inner product given by <var class="Arg">rep.innerproducts</var> is positive definite, otherwise returns false.</p>

<p><a id="X7CB2F52386F07B68" name="X7CB2F52386F07B68"></a></p>

<h5>5.2-3 MAJORANA_TestAxiomM2</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestAxiomM2</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the inner product given by <var class="Arg">rep.innerproducts</var> obeys axiom M2 (Norton's inequality), otherwise returns false.



<p><a id="X85DFEF167A236627" name="X85DFEF167A236627"></a></p>

<h5>5.2-4 MAJORANA_TestPrimitivity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestPrimitivity</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the 1-eigenspaces of all axes are 1-dimensional, otherwise returns false.</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">T := AsList( ConjugacyClass(G, (1,2)(3,4)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">input := ShapesOfMajoranaRepresentation(G,T);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep := MajoranaRepresentation(input, 2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NClosedMajoranaRepresentation(rep);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_IsComplete(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MajoranaAlgebraTest(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestFrobeniusForm(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestInnerProduct(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestAxiomM2(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestPrimitivity(rep);</span>
true
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap4.html">[Previous Chapter]</a>    <a href="chap6.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="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>

100%


¤ Dauer der Verarbeitung: 0.15 Sekunden  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




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.