Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/spinsym/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 1.9.2019 mit Größe 64 kB image not shown  

Quelle  chap2_mj.html   Sprache: HTML

 
 products/Sources/formale Sprachen/GAP/pkg/spinsym/doc/chap2_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 (SpinSym) - Chapter 2: Usage and features</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="chap2"  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="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="chap1_mj.html">[Previous Chapter]</a>    <a href="chapBib_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap2.html">[MathJax off]</a></p>
<p><a id="X8243772F820CC34E" name="X8243772F820CC34E"></a></p>
<div class="ChapSects"><a href="chap2_mj.html#X8243772F820CC34E">2 <span class="Heading">Usage and features</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7B57D3B284BE53F9">2.1 <span class="Heading">Accessing the tables</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7FFBFA7E7E31EA14">2.2 <span class="Heading">Character parameters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X87BE0EFA878CA593">2.2-1 <span class="Heading">Parameters of ordinary characters</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X85178CBC7E7039FF">2.2-2 <span class="Heading">Parameters of modular characters</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X857C89397E32A4E1">2.3 <span class="Heading">Class parameters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7B846F3782B35EF3">2.3-1 SpinSymStandardRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X78925C167ABA9462">2.3-2 SpinSymStandardRepresentativeImage</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7C885CF87A06E0DF">2.3-3 SpinSymPreimage</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8243866286DA609C">2.3-4 SpinSymBrauerCharacter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X87EF85BC816651BF">2.3-5 SpinSymBasicCharacter</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7BB8AC1D781BD6FE">2.4 <span class="Heading">Young subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X798B9B1482CBDDC0">2.4-1 SpinSymCharacterTableOfMaximalYoungSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X849A37657F53BAE3">2.4-2 SpinSymBrauerTableOfMaximalYoungSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7906869F7F190E76">2.5 <span class="Heading">Class Fusions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7C4BE18086E671CD">2.5-1 SpinSymClassFusion</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X857FB2A17904A9CE">2.5-2 SpinSymClassFusion2Ain2S</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X84DDBEEF7BA42414">2.5-3 SpinSymClassFusion2Sin2S</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A4425B58361A2FD">2.5-4 SpinSymClassFusion2Ain2A</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7BE629FB81C12F27">2.5-5 SpinSymClassFusion2Sin2A</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8310E6BC7CC1A2F5">2.5-6 SpinSymClassFusion2SSin2S</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7854E1E785A0378A">2.5-7 SpinSymClassFusion2SAin2SS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X83B113007DF2BB69">2.5-8 SpinSymClassFusion2ASin2SS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X786B6E2D7CF13B51">2.5-9 SpinSymClassFusion2AAin2SA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X80DFBE347B87AA43">2.5-10 SpinSymClassFusion2AAin2AS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X796897DF78143E05">2.5-11 SpinSymClassFusion2AAin2A</a></span>
</div></div>
</div>

<h3>2 <span class="Heading">Usage and features</span></h3>

<p><a id="X7B57D3B284BE53F9" name="X7B57D3B284BE53F9"></a></p>

<h4>2.1 <span class="Heading">Accessing the tables</span></h4>

<p>All Brauer tables in this package are relative to a <em>generic</em> ordinary character table obtained by one of the following constructions</p>


<dl>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "2.Sym(n)" )</code>, the character table of <span class="SimpleMath">\(2.Sym(n)\)</span> ,</p>

</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "2.Alt(n)" )</code>, the character table of <span class="SimpleMath">\(2.Alt(n)\)</span> ,</p>

</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "Sym(n)" )</code>, the character table of <span class="SimpleMath">\(Sym(n)\)</span>,</p>

</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "Alt(n)" )</code>, the character table of <span class="SimpleMath">\(Alt(n)\)</span>.</p>

</dd>
</dl>
<p>Note that these are synonymous expressions for</p>


<dl>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "DoubleCoverSymmetric", n )</code>,</p>

</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "DoubleCoverAlternating", n )</code>,</p>

</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "Symmetric", n )</code>,</p>

</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">CharacterTable( "Alternating", n )</code>,</p>

</dd>
</dl>
<p>respectively. More detailed information on these tables is to be found in <a href="chapBib_mj.html#biBNoeske2002">[3]</a>. In this manual, we call such a character table an (ordinary) <em>SpinSym table</em>. If <code class="code">ordtbl</code> is an ordinary SpinSym table, the relative Brauer table in characteristic <code class="code">p</code> can be accessed using the <code class="code">mod</code>-operator (i.e. <code class="code">ordtbl mod p;</code>). Such a Brauer table is called a (<span class="SimpleMath">\(p\)</span>-modular) <em>SpinSym table</em> in the following.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ordtbl:= CharacterTable( "2.Sym(18)" );</span>
CharacterTable( "2.Sym(18)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">modtbl:= ordtbl mod 3;</span>
BrauerTable( "2.Sym(18)", 3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdinaryCharacterTable(modtbl)=ordtbl;</span>
true

</pre></div>

<p><a id="X7FFBFA7E7E31EA14" name="X7FFBFA7E7E31EA14"></a></p>

<h4>2.2 <span class="Heading">Character parameters</span></h4>

<p>An ordinary SpinSym table has character parameters, that is, a list of suitable labels corresponding to the rows of <code class="code">ordtbl</code> and therefore the irreducible ordinary characters of the underlying group. See <code class="code">CharacterParameters()</code> in the GAP Reference Manual.</p>

<p><a id="X87BE0EFA878CA593" name="X87BE0EFA878CA593"></a></p>

<h5>2.2-1 <span class="Heading">Parameters of ordinary characters</span></h5>

<p>In the following, `ordinary (spin) character' is used synonymously for `irreducible ordinary (spin) character'. It is well known that there is a bijection between the set of ordinary characters of <span class="SimpleMath">\(Sym(n)\)</span> and the set <span class="SimpleMath">\(P(n)\)</span> of all partitions of <span class="SimpleMath">\(n\)</span>. Recall that a partition of a natural number <span class="SimpleMath">\(n\)</span> is a list of non-increasing positive integers (its <em>parts</em>) that sum up to <span class="SimpleMath">\(n\)</span>. In this way, every ordinary character <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(Sym(n)\)</span> has a label of the form <code class="code">[1,c]</code> where <code class="code">c</code> is a partition of <span class="SimpleMath">\(n\)</span>. The labels of the ordinary characters of <span class="SimpleMath">\(Alt(n)\)</span> are induced by Clifford theory as follows. Either the restriction <span class="SimpleMath">\(\psi=\chi|_{Alt(n)}\)</span> of <span class="SimpleMath">\(\chi\)</span> to <span class="SimpleMath">\(Alt(n)\)</span> is an ordinary character of <span class="SimpleMath">\(Alt(n)\)</span>, or <span class="SimpleMath">\(\psi\)</span> decomposes as the sum of two distinct ordinary characters <span class="SimpleMath">\(\psi_1\)</span> and <span class="SimpleMath">\(\psi_2\)</span>.</p>

<p>In the first case there is another ordinary character of <span class="SimpleMath">\(Sym(n)\)</span>, say <span class="SimpleMath">\(\xi\)</span> labelled by <code class="code">[1,d]</code>, such that the restriction of <span class="SimpleMath">\(\xi\)</span> to <span class="SimpleMath">\(Alt(n)\)</span> is equal to <span class="SimpleMath">\(\psi\)</span>. Moreover, the induced character of <span class="SimpleMath">\(Sym(n)\)</span> obtained from <span class="SimpleMath">\(\psi\)</span> decomposes as the sum of <span class="SimpleMath">\(\chi\)</span> and <span class="SimpleMath">\(\xi\)</span>. Then <span class="SimpleMath">\(\psi\)</span> is labelled by <code class="code">[1,c]</code> or <code class="code">[1,d]</code>.</p>

<p>In the second case, both <span class="SimpleMath">\(\psi_1\)</span> and <span class="SimpleMath">\(\psi_2\)</span> induce irreducibly up to <span class="SimpleMath">\(\chi\)</span>. Then <span class="SimpleMath">\(\psi_1\)</span> and <span class="SimpleMath">\(\psi_2\)</span> are labelled by <code class="code">[1,[c,'+']]</code> and <code class="code">[1,[c,'-']]</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ctS:= CharacterTable( "Sym(5)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterParameters(ctS);</span>
[ [ 1, [ 1, 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1, 1 ] ], [ 1, [ 2, 2, 1 ] ], 
  [ 1, [ 3, 1, 1 ] ], [ 1, [ 3, 2 ] ], [ 1, [ 4, 1 ] ], [ 1, [ 5 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ctA:= CharacterTable( "Alt(5)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterParameters(ctA);</span>
[ [ 1, [ 1, 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1, 1 ] ], [ 1, [ 2, 2, 1 ] ], 
  [ 1, [ [ 3, 1, 1 ], '+' ] ], [ 1, [ [ 3, 1, 1 ], '-' ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">chi:= Irr(ctS)[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi:= RestrictedClassFunction(chi,ctA);;           </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Position(Irr(ctA),psi);    </span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">xi:= Irr(ctS)[7];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RestrictedClassFunction(xi,ctA) = psi;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">InducedClassFunction(psi,ctS) = chi + xi;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">chi:= Irr(ctS)[4];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi:= RestrictedClassFunction(chi,ctA);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi1:= Irr(ctA)[4];; psi2:= Irr(ctA)[5];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi = psi1 + psi2;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">InducedClassFunction(psi1,ctS) = chi;              </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">InducedClassFunction(psi2,ctS) = chi;</span>
true

</pre></div>

<p>If <span class="SimpleMath">\(\chi\)</span> is an ordinary character of <span class="SimpleMath">\(2.Sym(n)\)</span> or <span class="SimpleMath">\(2.Alt(n)\)</span> , then <span class="SimpleMath">\(\chi(z)=\chi(1)\)</span> or <span class="SimpleMath">\(\chi(z)=-\chi(1)\)</span>. If <span class="SimpleMath">\(\chi(z)=\chi(1)\)</span>, then <span class="SimpleMath">\(\chi\)</span> is obtained by inflation (along the central subgroup generated by <span class="SimpleMath">\(z\)</span>) from an ordinary character of <span class="SimpleMath">\(Sym(n)\)</span> or <span class="SimpleMath">\(Alt(n)\)</span>, respectively, whose label is given to <span class="SimpleMath">\(\chi\)</span>. Otherwise, if <span class="SimpleMath">\(\chi\)</span> is a spin character, that is <span class="SimpleMath">\(\chi(z)=-\chi(1)\)</span>, then its label is described next.</p>

<p>The set of ordinary spin characters of <span class="SimpleMath">\(2.Sym(n)\)</span> is parameterized by the subset <span class="SimpleMath">\(D(n)\)</span> of <span class="SimpleMath">\(P(n)\)</span> of all distinct-parts partitions of <span class="SimpleMath">\(n\)</span> (also called bar partitions). If <code class="code">c</code> is an even distinct-parts partition of <span class="SimpleMath">\(n\)</span>, then there is a unique ordinary spin character of <span class="SimpleMath">\(2.Sym(n)\)</span> that is labelled by <code class="code">[2,c]</code>. In contrast, if <code class="code">c</code> is an odd distinct-parts partition of <span class="SimpleMath">\(n\)</span>, then there are two distinct ordinary spin characters of <span class="SimpleMath">\(2.Sym(n)\)</span> that are labelled by <code class="code">[2,[c,'+']]</code> and <code class="code">[2,[c,'-']]</code>. Now the labels of the ordinary spin characters of <span class="SimpleMath">\(2.Alt(n)\)</span> follow from the labels of <span class="SimpleMath">\(2.Sym(n)\)</span> in the same way as those of <span class="SimpleMath">\(Alt(n)\)</span> follow from the labels of <span class="SimpleMath">\(Sym(n)\)</span> (see the beginning of this subsection <a href="chap2_mj.html#X87BE0EFA878CA593"><span class="RefLink">2.2-1</span></a>).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ctS:= CharacterTable( "Sym(5)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ct2S:= CharacterTable( "2.Sym(5)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ch:= CharacterParameters(ct2S);</span>
[ [ 1, [ 1, 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1, 1 ] ], [ 1, [ 2, 2, 1 ] ], 
  [ 1, [ 3, 1, 1 ] ], [ 1, [ 3, 2 ] ], [ 1, [ 4, 1 ] ], [ 1, [ 5 ] ], 
  [ 2, [ [ 3, 2 ], '+' ] ], [ 2, [ [ 3, 2 ], '-' ] ], 
  [ 2, [ [ 4, 1 ], '+' ] ], [ 2, [ [ 4, 1 ], '-' ] ], [ 2, [ 5 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= Positions( List(ch, x-> x[1]), 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RestrictedClassFunctions( Irr(ctS), ct2S ) = Irr(ct2S){pos}; #inflation</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ct2A:= CharacterTable( "2.Alt(5)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterParameters(ct2A);</span>
[ [ 1, [ 1, 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1, 1 ] ], [ 1, [ 2, 2, 1 ] ], 
  [ 1, [ [ 3, 1, 1 ], '+' ] ], [ 1, [ [ 3, 1, 1 ], '-' ] ], [ 2, [ 3, 2 ] ], 
  [ 2, [ 4, 1 ] ], [ 2, [ [ 5 ], '+' ] ], [ 2, [ [ 5 ], '-' ] ] ]

</pre></div>

<p><a id="X85178CBC7E7039FF" name="X85178CBC7E7039FF"></a></p>

<h5>2.2-2 <span class="Heading">Parameters of modular characters</span></h5>

<p>In the following, `<span class="SimpleMath">\(p\)</span>-modular (spin) character' is used synonymously for `irreducible \(p\)-modular (spin) character'. The set of <span class="SimpleMath">\(p\)</span>-modular characters of <span class="SimpleMath">\(Sym(n)\)</span> is parameterized by the set of all <span class="SimpleMath">\(p\)</span>-regular partitions of <span class="SimpleMath">\(n\)</span>. A partition is <span class="SimpleMath">\(p\)</span>-regular if no part is repeated more than <span class="SimpleMath">\(p-1\)</span> times. Now every <span class="SimpleMath">\(p\)</span>-modular character <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(Sym(n)\)</span> has a label of the form <code class="code">[1,c]</code> where <code class="code">c</code> is a <span class="SimpleMath">\(p\)</span>-regular partition of <span class="SimpleMath">\(n\)</span>.</p>

<p>Again, the labels for the <span class="SimpleMath">\(p\)</span>-modular spin characters of <span class="SimpleMath">\(Alt(n)\)</span> follow from the labels of <span class="SimpleMath">\(Sym(n)\)</span>. However, comparing subsection <a href="chap2_mj.html#X87BE0EFA878CA593"><span class="RefLink">2.2-1</span></a>, their format is slightly different.</p>

<p>If <span class="SimpleMath">\(\chi\)</span> and <span class="SimpleMath">\(\xi\)</span> are distinct <span class="SimpleMath">\(p\)</span>-modular characters of <span class="SimpleMath">\(Sym(n)\)</span> that restrict to the same <span class="SimpleMath">\(p\)</span>-modular character <span class="SimpleMath">\(\psi\)</span> of <span class="SimpleMath">\(Alt(n)\)</span>, then <span class="SimpleMath">\(\psi\)</span> is labelled by <code class="code">[1,[c,'0']]</code> where either <span class="SimpleMath">\(\chi\)</span> or <span class="SimpleMath">\(\xi\)</span> is labelled by <code class="code">[1,c]</code>. If <span class="SimpleMath">\(\chi\)</span> is a <span class="SimpleMath">\(p\)</span>-modular character of <span class="SimpleMath">\(Sym(n)\)</span> whose restriction to <span class="SimpleMath">\(Alt(n)\)</span> decomposes as the sum of two distinct <span class="SimpleMath">\(p\)</span>-modular characters, then these are labelled by <code class="code">[1,[c,'+']]</code> and <code class="code">[1,[c,'-']]</code> where <span class="SimpleMath">\(\chi\)</span> is labelled by <code class="code">[1,c]</code>.</p>

<p>As in the ordinary case, the set of <span class="SimpleMath">\(p\)</span>-modular characters of <span class="SimpleMath">\(2.Sym(n)\)</span> is the union of the subset consisting of all inflated <span class="SimpleMath">\(p\)</span>-modular characters of <span class="SimpleMath">\(Sym(n)\)</span> and the subset of spin characters characterized by negative integer values on the central element <span class="SimpleMath">\(z\)</span>. The analogue statement holds for <span class="SimpleMath">\(2.Alt(n)\)</span>. The set of <span class="SimpleMath">\(p\)</span>-modular spin characters of <span class="SimpleMath">\(2.Sym(n)\)</span> is parameterized by the set of all restricted <span class="SimpleMath">\(p\)</span>-strict partitions of <span class="SimpleMath">\(n\)</span>. A partition is called <span class="SimpleMath">\(p\)</span>-strict if every repeated part is divisible by <span class="SimpleMath">\(p\)</span>, and a <span class="SimpleMath">\(p\)</span>-strict partition <span class="SimpleMath">\(\lambda\)</span> is restricted if <span class="SimpleMath">\(\lambda_i-\lambda_{i+1}<p\)</span> whenever <span class="SimpleMath">\(\lambda_i\)</span> is divisible <span class="SimpleMath">\(p\)</span>, and <span class="SimpleMath">\(\lambda_i-\lambda_{i+1}\leq p\)</span> otherwise for all parts <span class="SimpleMath">\(\lambda_i\)</span> of <span class="SimpleMath">\(\lambda\)</span> (where we set <span class="SimpleMath">\(\lambda_{i+1}=0\)</span> if <span class="SimpleMath">\(\lambda_i\)</span> is the last part). If <code class="code">c</code> is a restricted <span class="SimpleMath">\(p\)</span>-strict partition of <span class="SimpleMath">\(n\)</span> such that <span class="SimpleMath">\(n\)</span> minus the number of parts not divisible by <span class="SimpleMath">\(p\)</span> is even, then there is a unique <span class="SimpleMath">\(p\)</span>-modular spin character of <span class="SimpleMath">\(2.Sym(n)\)</span> that is labelled by <code class="code">[2,[c,'0']]</code>. Its restriction to <span class="SimpleMath">\(2.Alt(n)\)</span> decomposes as the sum of two distinct <span class="SimpleMath">\(p\)</span>-modular characters which are labelled by <code class="code">[2,[c,'+']]</code> and <code class="code">[2,[c,'-']]</code>. If <span class="SimpleMath">\(n\)</span> minus the number of parts of <code class="code">c</code> that are not divisible by <span class="SimpleMath">\(p\)</span> is odd, then there are two distinct <span class="SimpleMath">\(p\)</span>-modular spin characters of <span class="SimpleMath">\(2.Sym(n)\)</span> that are labelled by <code class="code">[2,[c,'+']]</code> and <code class="code">[2,[c,'-']]</code>. Both of these characters restrict to the same irreducible <span class="SimpleMath">\(p\)</span>-modular spin character of <span class="SimpleMath">\(2.Alt(n)\)</span> which is labelled by <code class="code">[2,[c,'0']]</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ctS:= CharacterTable( "Sym(5)" ) mod 3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ct2S:= CharacterTable( "2.Sym(5)" ) mod 3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ch:= CharacterParameters(ct2S);</span>
[ [ 1, [ 5 ] ], [ 1, [ 4, 1 ] ], [ 1, [ 3, 2 ] ], 
  [ 1, [ 3, 1, 1 ] ], [ 1, [ 2, 2, 1 ] ], 
  [ 2, [ [ 4, 1 ], '+' ] ], [ 2, [ [ 4, 1 ], '-' ] ], 
  [ 2, [ [ 3, 2 ], '0' ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= Positions( List(ch, x-> x[1]), 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RestrictedClassFunctions( Irr(ctS), ct2S ) = Irr(ct2S){pos}; #inflation</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ct2A:= CharacterTable( "2.Alt(5)" ) mod 3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterParameters(ct2A);</span>
[ [ 1, [ [ 5 ], '0' ] ], [ 1, [ [ 4, 1 ], '0' ] ], 
  [ 1, [ [ 3, 1, 1 ], '+' ] ], [ 1, [ [ 3, 1, 1 ], '-' ] ], 
  [ 2, [ [ 4, 1 ], '0' ] ], [ 2, [ [ 3, 2 ], '+' ] ], [ 2, [ [ 3, 2 ], '-' ] ] ]

</pre></div>

<p><a id="X857C89397E32A4E1" name="X857C89397E32A4E1"></a></p>

<h4>2.3 <span class="Heading">Class parameters</span></h4>

<p>Let <code class="code">ct</code> be an ordinary SpinSym table. Then <code class="code">ct</code> has a list of class parameters, that is, a list of suitable labels corresponding to the columns of <code class="code">ct</code> and therefore the conjugacy classes of the underlying group. See <code class="code">ClassParameters()</code> in the GAP Reference Manual. If <code class="code">bt</code> is a Brauer table in characteristic <span class="SimpleMath">\(p\)</span> relative to <code class="code">ct</code>, its class parameters are inherited from <code class="code">ct</code> in correspondence with the <span class="SimpleMath">\(p\)</span>-regular conjugacy classes of the underlying group.</p>

<p>Let <span class="SimpleMath">\(P(n)\)</span> denote the set of partitions of <span class="SimpleMath">\(n\)</span>.</p>

<p>The conjugacy classes of <span class="SimpleMath">\(Sym(n)\)</span> are naturally parameterized by the cycle types of their elements, and each cycle type corresponds to a partition of <span class="SimpleMath">\(n\)</span>. Therefore a conjugacy class <span class="SimpleMath">\(C\)</span> of <span class="SimpleMath">\(Sym(n)\)</span> is characterized by its <em>type</em> <span class="SimpleMath">\(c\)</span> in <span class="SimpleMath">\(P(n)\)</span>. The corresponding entry in the list of class parameters is <code class="code">[1,c]</code>. Assume that <span class="SimpleMath">\(C\)</span> is a subset of <span class="SimpleMath">\(Alt(n)\)</span>. Then <span class="SimpleMath">\(C\)</span> is also a conjugacy class of <span class="SimpleMath">\(Alt(n)\)</span> if and only if not all parts of <span class="SimpleMath">\(c\)</span> are odd and pairwise distinct. Otherwise, <span class="SimpleMath">\(C\)</span> splits as the union of two distinct <span class="SimpleMath">\(Alt(n)\)</span>-classes of the same size, <span class="SimpleMath">\(C^+\)</span> of type <span class="SimpleMath">\(c^+\)</span> and <span class="SimpleMath">\(C^-\)</span> of type <span class="SimpleMath">\(c^-\)</span>. The corresponding entries in the list of class parameters are <code class="code">[1,[c,'+']]</code> and <code class="code">[1,[c,'-']]</code>, respectively.</p>

<p>Furthermore, the preimage <span class="SimpleMath">\(C'=C^{{\pi^-1}}\) is either a conjugacy class of \(2.Sym(n)\) of type \(c\) with class parameter [1,c], or \(C'\)</span> splits as the union of two distinct <span class="SimpleMath">\(2.Sym(n)\)</span> -classes <span class="SimpleMath">\(C'_1\) and \(C'_2=zC'_1\) , both of type \(c\) with corresponding class parameters [1,c] and [2,c], respectively. An analogous description applies for the conjugacy classes of \(2.Alt(n)\) .




<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ct:= CharacterTable( "Sym(3)" );;  </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters(ct);</span>
[ [ 1, [ 1, 1, 1 ] ], [ 1, [ 2, 1 ] ], [ 1, [ 3 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ct:= CharacterTable( "Alt(3)" );;  </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters(ct);</span>
[ [ 1, [ 1, 1, 1 ] ], [ 1, [ [ 3 ], '+' ] ], [ 1, [ [ 3 ], '-' ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ct:= CharacterTable( "2.Sym(3)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters(ct);</span>
[ [ 1, [ 1, 1, 1 ] ], [ 2, [ 1, 1, 1 ] ], [ 1, [ 2, 1 ] ], [ 2, [ 2, 1 ] ], 
  [ 1, [ 3 ] ], [ 2, [ 3 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ct:= CharacterTable( "2.Alt(3)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters(ct);</span>
[ [ 1, [ 1, 1, 1 ] ], [ 2, [ 1, 1, 1 ] ], 
  [ 1, [ [ 3 ], '+' ] ], [ 2, [ [ 3 ], '+' ] ], 
  [ 1, [ [ 3 ], '-' ] ], [ 2, [ [ 3 ], '-' ] ] ]

</pre></div>

<p>To each conjugacy class of <span class="SimpleMath">\(2.Sym(n)\)</span> or <span class="SimpleMath">\(2.Alt(n)\)</span> a certain standard representative is assigned in the following way. Let <span class="SimpleMath">\(c=[c_1,c_2,\ldots,c_m]\)</span> be a partition of <span class="SimpleMath">\(n\)</span>. We set <span class="SimpleMath">\(d_1=0\)</span>, <span class="SimpleMath">\(d_i=c_1+\ldots +c_{i-1}\)</span> for <span class="SimpleMath">\(i\geq 2\)</span>, and</p>

<p class="center">\[t(c_i,d_i)= t_{d_i+1}t_{d_i+2}\ldots t_{d_i+c_i-1}\]</p>

<p>for <span class="SimpleMath">\(1\leq i\leq m-1\)</span>, where <span class="SimpleMath">\(t(c_i,d_i)= 1\)</span> if <span class="SimpleMath">\(c_i=1\)</span>. The <em>standard representative of type</em> <span class="SimpleMath">\(c\)</span> is defined as</p>

<p class="center">\[t_c=t(c_1,d_1)t(c_2,d_2)...t(c_{m-1},d_{m-1}).\]</p>

<p>Furthermore, we define the standard representatives of type <span class="SimpleMath">\(c^+=\)</span><code class="code">[c,'+']</code> and <span class="SimpleMath">\(c^-=\)</span><code class="code">[c,'-']</code> to be <span class="SimpleMath">\(t_{c^+}=t_{c}\)</span> and <span class="SimpleMath">\(t_{c^-}=t_1^{-1}t_c t_1 \)</span>, respectively.</p>

<p>For example, the standard representative of type <span class="SimpleMath">\(c=[7,4,3,1]\)</span> in <span class="SimpleMath">\(P(15)\)</span> is</p>

<p class="center">\[t_c=t_1t_2t_3t_4t_5t_6t_8t_9t_{10}t_{12}t_{13}.\]</p>

<p>Now <span class="SimpleMath">\(C'\) is a conjugacy class of \(2.Sym(n)\) or \(2.Alt(n)\) with parameter




<dl>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">[1,c]</code> if and only if <span class="SimpleMath">\(t_c \)</span> is an element of <span class="SimpleMath">\(C'\) ,



</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">[2,c]</code> if and only if <span class="SimpleMath">\(zt_c\)</span> is an element of <span class="SimpleMath">\(C'\) ,



</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">[1,[c,'+']]</code> if and only if <span class="SimpleMath">\(t_{c^+}\)</span> is an element of <span class="SimpleMath">\(C'\) ,



</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">[2,[c,'+']]</code> if and only if <span class="SimpleMath">\(zt_{c^+}\)</span> is an element of <span class="SimpleMath">\(C'\) ,



</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">[1,[c,'-']]</code> if and only if <span class="SimpleMath">\(t_{c^-}\)</span> is an element of <span class="SimpleMath">\(C'\) ,



</dd>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">[2,[c,'-']]</code> if and only if <span class="SimpleMath">\(zt_{c^-}\)</span> is an element of <span class="SimpleMath">\(C'\) .



</dd>
</dl>
<p><a id="X7B846F3782B35EF3" name="X7B846F3782B35EF3"></a></p>

<h5>2.3-1 SpinSymStandardRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymStandardRepresentative</code>( <var class="Arg">c</var>, <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the image of the standard representative of type <var class="Arg">c</var> under a given <span class="SimpleMath">\(2.Sym(n)\)</span>-representation.</p>

<p>Expecting the second entry of a class parameter of <span class="SimpleMath">\(2.Sym(n)\)</span> or <span class="SimpleMath">\(2.Alt(n)\)</span> , say <var class="Arg">c</var>, the standard representative of type <var class="Arg">c</var> under a given representation of <span class="SimpleMath">\(2.Sym(n)\)</span> is computed. The argument <var class="Arg">rep</var> is assumed to be a list <span class="SimpleMath">\([t_1^R,t_2^R,\ldots,t_{n-1}^R]\)</span> given by the images of the generators <span class="SimpleMath">\(t_1,\ldots,t_{n-1}\)</span> of <span class="SimpleMath">\(2.Sym(n)\)</span> under a (not necessarily faithful) representation <span class="SimpleMath">\(R\)</span> of <span class="SimpleMath">\(2.Sym(n)\)</span> .</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ct:= CharacterTable("2.Sym(15)") mod 5;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cl:= ClassParameters(ct)[99];</span>
[ 1, [ 7, 4, 3, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= cl[2];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep:= BasicSpinRepresentationOfSymmetricGroup(15,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= SpinSymStandardRepresentative(c,rep); </span>
< immutable compressed matrix 64x64 over GF(25) >
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives(ct)[99];</span>
168
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(t);</span>
168
<span class="GAPprompt">gap></span> <span class="GAPinput">BrauerCharacterValue(t);</span>
0

</pre></div>

<p><a id="X78925C167ABA9462" name="X78925C167ABA9462"></a></p>

<h5>2.3-2 SpinSymStandardRepresentativeImage</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymStandardRepresentativeImage</code>( <var class="Arg">c</var>[, <var class="Arg">j</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the image of the standard representative of type <var class="Arg">c</var> under the natural epimorphism <span class="SimpleMath">\(\pi:2.Sym({{j,...,j+n-1}}) \to Sym({{j,...,j+n-1}})\)</span>.</p>

<p>Given the second entry <var class="Arg">c</var> of a class parameter of <span class="SimpleMath">\(2.Sym(n)\)</span> or <span class="SimpleMath">\(2.Alt(n)\)</span> , and optionally a positive integer <var class="Arg">j</var>, the image of the standard representative of type <var class="Arg">c</var> under <span class="SimpleMath">\(\pi:2.Sym({{j,...,j+n-1}}) \to Sym({{j,...,j+n-1}})\)</span> with <span class="SimpleMath">\(t_i^\pi=(i,i+1)\)</span> for <span class="SimpleMath">\(j\leq i\leq j+n-2\)</span> is computed by calling <code class="keyw">SpinSymStandardRepresentative(c,rep)</code> where <code class="code">rep</code> is the list <code class="code">[(j,j+1),(j+1,j+2),...,(j+n-2,j+n-1)]</code>. By default, <code class="code">j=1</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">s1:= SpinSymStandardRepresentativeImage([7,4,3,1]);         </span>
(1,7,6,5,4,3,2)(8,11,10,9)(12,14,13)
<span class="GAPprompt">gap></span> <span class="GAPinput">s2:= SpinSymStandardRepresentativeImage([[7,4,3,1],'-']);</span>
(1,2,7,6,5,4,3)(8,11,10,9)(12,14,13)
<span class="GAPprompt">gap></span> <span class="GAPinput">s2 = s1^(1,2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SpinSymStandardRepresentativeImage([7,4,3,1],3);       </span>
(3,9,8,7,6,5,4)(10,13,12,11)(14,16,15)

</pre></div>

<p><a id="X7C885CF87A06E0DF" name="X7C885CF87A06E0DF"></a></p>

<h5>2.3-3 SpinSymPreimage</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymPreimage</code>( <var class="Arg">c</var>, <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a (standard) lift of the element <var class="Arg">c</var> of <span class="SimpleMath">\(Sym(n)\)</span> in <span class="SimpleMath">\(2.Sym(n)\)</span> under a given <span class="SimpleMath">\(2.Sym(n)\)</span>-representation.</p>

<p>See <a href="chapBib_mj.html#biBMaas2011">[2, (5.1.12)]</a> for the definition of the lift that is returned by this function. The permutation <var class="Arg">c</var> is written as a product of simple transpositions <span class="SimpleMath">\((i,i+1)\)</span>, then these are replaced by the images of their canonical lifts <span class="SimpleMath">\(t_i\)</span> under a given representation <span class="SimpleMath">\(R\)</span> of <span class="SimpleMath">\(2.Sym(n)\)</span> (recall the beginning of Chapter <a href="chap1_mj.html#X7DFB63A97E67C0A1"><span class="RefLink">1</span></a> for the definition of <span class="SimpleMath">\(t_i\)</span>). Here <var class="Arg">rep</var> is assumed to be the list <span class="SimpleMath">\([t_1^R,t_2^R,\ldots,t_{n-1}^R]\)</span>.</p>

<p>Note that a more efficient computation may be achieved by computing and storing a list of all necessary transpositions once and for all, before lifting (many) elements (under a possibly large representation).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">rep:= BasicSpinRepresentationOfSymmetricGroup(15);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= SpinSymStandardRepresentativeImage([5,4,3,2,1]);</span>
(1,5,4,3,2)(6,9,8,7)(10,12,11)(13,14)
<span class="GAPprompt">gap></span> <span class="GAPinput">C:= SpinSymPreimage(c,rep);</span>
< immutable compressed matrix 64x64 over GF(9) >
<span class="GAPprompt">gap></span> <span class="GAPinput">C = SpinSymStandardRepresentative([5,4,3,2,1],rep);</span>
true

</pre></div>

<p><a id="X8243866286DA609C" name="X8243866286DA609C"></a></p>

<h5>2.3-4 SpinSymBrauerCharacter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymBrauerCharacter</code>( <var class="Arg">ccl</var>, <var class="Arg">ords</var>, <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the Brauer character afforded by a given representation of <span class="SimpleMath">\(2.Sym(n)\)</span> .</p>

<p>This function is based on a simplified computation of the <strong class="pkg">GAP</strong> attribute <code class="code">BrauerCharacterValue(mat)</code> for an invertible matrix <code class="code">mat</code> over a finite field whose characteristic is coprime to the order of <code class="code">mat</code>.</p>

<p>The arguments <var class="Arg">ccl</var> and <var class="Arg">ords</var> are expected to be the values of the attributes <code class="code">ClassParameters(modtbl)</code> and <code class="code">OrdersClassRepresentatives(modtbl)</code> of a (possibly incomplete) <span class="SimpleMath">\(p\)</span>-modular SpinSym table <code class="code">modtbl</code> of <span class="SimpleMath">\(2.Sym(n)\)</span> .</p>

<p>The argument <var class="Arg">rep</var> is assumed to be a list <span class="SimpleMath">\([t_1^R,t_2^R,\ldots,t_{n-1}^R]\)</span> given by the images of the generators <span class="SimpleMath">\(t_1,\ldots,t_{n-1}\)</span> of <span class="SimpleMath">\(2.Sym(n)\)</span> under a (not necessarily faithful) <span class="SimpleMath">\(2.Sym(n)\)</span> -representation <span class="SimpleMath">\(R\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ct:= CharacterTable("DoubleCoverSymmetric",15);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">bt:= CharacterTableRegular(ct,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= GetFusionMap(bt,ct);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ccl:= ClassParameters(ct){fus};;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ords:= OrdersClassRepresentatives(bt);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep:= BasicSpinRepresentationOfSymmetricGroup(15,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi:= SpinSymBrauerCharacter(ccl,ords,rep);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi in Irr(ct mod 5);</span>
true

</pre></div>

<p><a id="X87EF85BC816651BF" name="X87EF85BC816651BF"></a></p>

<h5>2.3-5 SpinSymBasicCharacter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymBasicCharacter</code>( <var class="Arg">modtbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a <span class="SimpleMath">\(p\)</span>-modular basic spin character of the (possibly incomplete) <span class="SimpleMath">\(p\)</span>-modular SpinSym table <var class="Arg">modtbl</var> of <span class="SimpleMath">\(2.Sym(n)\)</span> .</p>

<p>This is just a shortcut for constructing a basic spin representation of <span class="SimpleMath">\(2.Sym(n)\)</span> in characteristic <span class="SimpleMath">\(p\)</span> and computing its Brauer character by calling <code class="func">SpinSymBrauerCharacter</code> (<a href="chap2_mj.html#X8243866286DA609C"><span class="RefLink">2.3-4</span></a>) afterwards.</p>


<div class="example"><pre>
 
<span class="GAPprompt">gap></span> <span class="GAPinput">SetClassParameters(bt,ccl);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SpinSymBasicCharacter(bt) = phi;</span>
true

</pre></div>

<p><a id="X7BB8AC1D781BD6FE" name="X7BB8AC1D781BD6FE"></a></p>

<h4>2.4 <span class="Heading">Young subgroups</span></h4>

<p>Let <span class="SimpleMath">\(k\)</span> and <span class="SimpleMath">\(l\)</span> be integers greater than <span class="SimpleMath">\(1\)</span> and set <span class="SimpleMath">\(n=k+l\)</span>. The following subgroup of <span class="SimpleMath">\(2.Sym(n)\)</span> ,</p>

<p class="center">\[2.(Sym(k){\times}Sym(l)) = 
< t_1,...,t_{k-1}, t_{k+1},...,t_{n-1} >,\]</p>

<p>is called a (maximal) <em>Young subgroup</em> of <span class="SimpleMath">\(2.Sym(n)\)</span> . Similarly, the intersection <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> of <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span> and <span class="SimpleMath">\(2.Alt(n)\)</span> is a (maximal) Young subgroup of <span class="SimpleMath">\(2.Alt(n)\)</span> . Note that <span class="SimpleMath">\((2.(Sym(k){\times}Sym(l)))^\pi\)</span> is isomorphic to <span class="SimpleMath">\(Sym(k){\times}Sym(l)\)</span> and <span class="SimpleMath">\((2.(Alt(k){\times}Alt(l)))^\pi\)</span> is isomorphic to <span class="SimpleMath">\( Alt(k){\times}Alt(l)\)</span> but only <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> which is isomorphic to <span class="SimpleMath">\((2.Alt(k){\times}2.Alt(l))/ <(z,z)>\)</span> is a central product. In between <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span> there are further central products <span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span> which is isomorphic to <span class="SimpleMath">\((2.Sym(k){\times}2.Alt(l))/<(z,z)>\)</span> and <span class="SimpleMath">\(2.(Alt(k){\times}Sym(l)) \)</span> which is isomorphic to <span class="SimpleMath">\((2.Alt(k){\times}2.Sym(l))/<(z,z)>\)</span> which are <span class="SimpleMath">\(\pi\)</span>-preimages of <span class="SimpleMath">\(Sym(k){\times}Alt(l)\)</span> and <span class="SimpleMath">\(Alt(k){\times}Sym(l)\)</span>, respectively. See <a href="chapBib_mj.html#biBMaas2011">[2, Section 5.2]</a>.</p>

<p><a id="X798B9B1482CBDDC0" name="X798B9B1482CBDDC0"></a></p>

<h5>2.4-1 SpinSymCharacterTableOfMaximalYoungSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymCharacterTableOfMaximalYoungSubgroup</code>( <var class="Arg">k</var>, <var class="Arg">l</var>, <var class="Arg">type</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the ordinary character table of a maximal Young subgroup depending on <var class="Arg">type</var>.</p>

<p>For integers <var class="Arg">k</var> and <var class="Arg">l</var> greater than <span class="SimpleMath">\(1\)</span> the function returns the ordinary character table of <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span>, <span class="SimpleMath">\(2.(Alt(k){\times}Sym(l))\)</span>, <span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span>, or <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span> depending on the string <var class="Arg">type</var> being <code class="code">"Alternating"</code>, <code class="code">"AlternatingSymmetric"</code>, <code class="code">"SymmetricAlternating"</code>, or <code class="code">"Symmetric"</code>, respectively.</p>

<p>If <var class="Arg">type</var> is <code class="code">"Symmetric"</code> then the output is computed by means of Clifford's theory from the character tables of \(2.(Sym(k){\times}Alt(l))\), \(2.(Alt(k){\times}Alt(l))\), and \(2.(Alt(k){\times}Sym(l))\) (see [2, Section 5.2]). These `ingredients' are computed and then stored in the attribute <code class="code">SpinSymIngredients</code> so they can be accessed during the construction (and for the construction of a relative Brauer table too, see <code class="func">SpinSymBrauerTableOfMaximalYoungSubgroup</code> (<a href="chap2_mj.html#X849A37657F53BAE3"><span class="RefLink">2.4-2</span></a>)).</p>

<p>The construction of the character tables of <var class="Arg">type</var> <code class="code">"Alternating"</code>, <code class="code">"AlternatingSymmetric"</code>, or <code class="code">"SymmetricAlternating"</code> is straightforward and may be accomplished by first construcing a direct product, for example, the character table of <span class="SimpleMath">\(2.Sym(k){\times}2.Alt(l)\)</span>, followed by the construction of the character table of the factor group mod <span class="SimpleMath">\(<(z,z)>\)</span>.</p>

<p>However, we use a faster method that builds up the table from scratch, using the appropriate component tables as ingredients (for example, the generic character tables of <span class="SimpleMath">\(2.Sym(k)\)</span> and <span class="SimpleMath">\(2.Alt(l)\)</span> ). In this way we can easily build up a suitable list of class parameters that are needed to determine the class fusion in the construction of <var class="Arg">type</var> <code class="code">"Symmetric"</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">2AA:= SpinSymCharacterTableOfMaximalYoungSubgroup(8,5,"Alternating"); </span>
CharacterTable( "2.(Alt(8)xAlt(5))" )
<span class="GAPprompt">gap></span> <span class="GAPinput">SpinSymCharacterTableOfMaximalYoungSubgroup(8,5,"AlternatingSymmetric");</span>
CharacterTable( "2.(Alt(8)xSym(5))" )
<span class="GAPprompt">gap></span> <span class="GAPinput">SpinSymCharacterTableOfMaximalYoungSubgroup(8,5,"SymmetricAlternating");     </span>
CharacterTable( "2.(Sym(8)xAlt(5))" )
<span class="GAPprompt">gap></span> <span class="GAPinput">2SS:= SpinSymCharacterTableOfMaximalYoungSubgroup(8,5,"Symmetric");           </span>
CharacterTable( "2.(Sym(8)xSym(5))" )

</pre></div>

<p><a id="X849A37657F53BAE3" name="X849A37657F53BAE3"></a></p>

<h5>2.4-2 SpinSymBrauerTableOfMaximalYoungSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymBrauerTableOfMaximalYoungSubgroup</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the <var class="Arg">p</var>-modular character table of the ordinary character table <var class="Arg">ordtbl</var> returned by the function <code class="func">SpinSymCharacterTableOfMaximalYoungSubgroup</code> (<a href="chap2_mj.html#X798B9B1482CBDDC0"><span class="RefLink">2.4-1</span></a>).</p>

<p>If the rational prime <var class="Arg">p</var> is odd, then the construction of the irreducible Brauer characters is really the same as in the ordinary case but it depends on the <var class="Arg">p</var>-modular tables of of <var class="Arg">ordtbl</var>'s `ingredients'. If some Brauer table that is necessary for the construction is not available then <code class="code">fail</code> is returned.</p>

<p>Alternatively, the <code class="code">mod</code>-operator may be used.</p>

<p>For <var class="Arg">p</var> <span class="SimpleMath">\(=2\)</span> the Brauer table is essentially constructed as a direct product by standard <strong class="pkg">GAP</strong> methods written by Thomas Breuer.</p>

<p>We call a character table returned by <code class="func">SpinSymCharacterTableOfMaximalYoungSubgroup</code> (<a href="chap2_mj.html#X798B9B1482CBDDC0"><span class="RefLink">2.4-1</span></a>) or <code class="func">SpinSymBrauerTableOfMaximalYoungSubgroup</code> a SpinSym table too. It has lists of class and character parameters whose format is explained in <a href="chapBib_mj.html#biBMaas2011">[2, Sections 5.2, 5.3]</a>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">SpinSymBrauerTableOfMaximalYoungSubgroup(2AA,3);</span>
BrauerTable( "2.(Alt(8)xAlt(5))", 3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">2SS mod 5;</span>
BrauerTable( "2.(Sym(8)xSym(5))", 5 )
<span class="GAPprompt">gap></span> <span class="GAPinput">ct:= 2SS mod 2;  </span>
BrauerTable( "2.(Sym(8)xSym(5))", 2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">ct1:= CharacterTable("Sym(8)") mod 2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ct2:= CharacterTable("Sym(5)") mod 2;;     </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Irr(ct1*ct2) = Irr(ct);</span>
true

</pre></div>

<p><a id="X7906869F7F190E76" name="X7906869F7F190E76"></a></p>

<h4>2.5 <span class="Heading">Class Fusions</span></h4>

<p>The following functions determine class fusion maps between SpinSym tables by means of their class parameters. Such `default' class fusion maps allow to induce characters from various subgroups of \(2.Sym(n)\) or \(2.Alt(n)\) consistently.



<p><a id="X7C4BE18086E671CD" name="X7C4BE18086E671CD"></a></p>

<h5>2.5-1 SpinSymClassFusion</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion</code>( <var class="Arg">ctSource</var>, <var class="Arg">ctDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map from the SpinSym table <var class="Arg">ctSource</var> to the SpinSym table <var class="Arg">ctDest</var>. This map is stored if there is no other fusion map from <var class="Arg">ctSource</var> to <var class="Arg">ctDest</var> stored yet.</p>

<p>The possible input tables are expected to be either ordinary or <span class="SimpleMath">\(p\)</span>-modular SpinSym tables of the following pairs of groups</p>

<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdcenter">Source</td>
<td class="tdcenter"><span class="SimpleMath">\(\to\)</span></td>
<td class="tdcenter">Dest</td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.Alt(n)\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.Sym(n)\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.Sym(k)\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.Sym(n)\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.Alt(k)\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.Alt(n)\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.Sym(n-2)\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.Alt(n)\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.Sym(k+l)\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.(Alt(k){\times}Sym(l))\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.(Alt(k){\times}Sym(l))\)</span></td>
</tr>
<tr>
<td class="tdcenter"><span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span></td>
<td class="tdcenter"></td>
<td class="tdcenter"><span class="SimpleMath">\(2.Alt(k+l)\)</span></td>
</tr>
</table><br /><p> </p><br />
</div>

<p>The appropriate function (see the descriptions below) is called to determine the fusion map <code class="code">fus</code>. If <code class="code">GetFusionMap(ctSource, ctDest)</code> fails, then <code class="code">fus</code> is stored by calling <code class="code">StoreFusion(ctSource, fus, ctDest)</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ctD:= CharacterTable("2.Sym(18)");;                                  </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ctS:= SpinSymCharacterTableOfMaximalYoungSubgroup(10,8,"Symmetric");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap(ctS,ctD);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">SpinSymClassFusion(ctS,ctD);;</span>
#I SpinSymClassFusion: stored fusion map from 2.(Sym(10)xSym(8)) to 2.Sym(18)
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap(ctS,ctD) <> fail;</span>
true

</pre></div>

<p><a id="X857FB2A17904A9CE" name="X857FB2A17904A9CE"></a></p>

<h5>2.5-2 SpinSymClassFusion2Ain2S</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2Ain2S</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.Alt(n)\)</span> and <span class="SimpleMath">\(2.Sym(n)\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.Alt(n)\)</span> and <span class="SimpleMath">\(2.Sym(n)\)</span>, respectively, a corresponding class fusion map is determined. See <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.1)]</a>.</p>

<p><a id="X84DDBEEF7BA42414" name="X84DDBEEF7BA42414"></a></p>

<h5>2.5-3 SpinSymClassFusion2Sin2S</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2Sin2S</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.Sym(k)\)</span> and <span class="SimpleMath">\(2.Sym(n)\)</span> for <span class="SimpleMath">\(k\leq n\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.Sym(k)\)</span> and <span class="SimpleMath">\(2.Sym(n)\)</span> for <span class="SimpleMath">\(k\leq n\)</span>, respectively, a corresponding class fusion map is determined. See <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.2)]</a>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ctD:= CharacterTable("2.Sym(18)");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ctS:= CharacterTable("2.Sym(6)");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cclD:= ClassParameters(ctD);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cclS:= ClassParameters(ctS);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= SpinSymClassFusion2Sin2S(cclS,cclD);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StoreFusion(ctS,fus,ctD);</span>

</pre></div>

<p><a id="X7A4425B58361A2FD" name="X7A4425B58361A2FD"></a></p>

<h5>2.5-4 SpinSymClassFusion2Ain2A</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2Ain2A</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.Alt(k)\)</span> and <span class="SimpleMath">\(2.Alt(n)\)</span> for <span class="SimpleMath">\(k\leq n\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.Alt(k)\)</span> and <span class="SimpleMath">\(2.Alt(n)\)</span> for <span class="SimpleMath">\(k\leq n\)</span>, respectively, a corresponding class fusion map is determined. See <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.3)]</a>.</p>

<p><a id="X7BE629FB81C12F27" name="X7BE629FB81C12F27"></a></p>

<h5>2.5-5 SpinSymClassFusion2Sin2A</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2Sin2A</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.Sym(n-2)\)</span> and <span class="SimpleMath">\(2.Alt(n)\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.Sym(n-2)\)</span> and <span class="SimpleMath">\(2.Alt(n)\)</span>, respectively, a corresponding class fusion map with respect to the embedding of <span class="SimpleMath">\(< t_1t_{n-2},\ldots,t_{n-3}t_{n-1} >\)</span> isomorphic to <span class="SimpleMath">\(2.Sym(n-2)\)</span> in <span class="SimpleMath">\(2.Alt(n)\)</span> is determined. See <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.4)]</a>.</p>

<p><a id="X8310E6BC7CC1A2F5" name="X8310E6BC7CC1A2F5"></a></p>

<h5>2.5-6 SpinSymClassFusion2SSin2S</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2SSin2S</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span> and <span class="SimpleMath">\(2.Sym(k+l)\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span> and <span class="SimpleMath">\(2.Sym(k+l)\)</span>, respectively, a corresponding class fusion map is determined by means of <a href="chapBib_mj.html#biBMaas2011">[2, (5.1.6)]</a>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ctD:= CharacterTable("2.Sym(18)");;                                  </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ctS:= SpinSymCharacterTableOfMaximalYoungSubgroup(10,8,"Symmetric");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cclD:= ClassParameters(ctD);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cclS:= ClassParameters(ctS);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= SpinSymClassFusion2SSin2S(cclS,cclD);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StoreFusion(ctS,fus,ctD);</span>

</pre></div>

<p><a id="X7854E1E785A0378A" name="X7854E1E785A0378A"></a></p>

<h5>2.5-7 SpinSymClassFusion2SAin2SS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2SAin2SS</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span>, respectively, a corresponding class fusion map is determined. See <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.6)]</a>.</p>

<p><a id="X83B113007DF2BB69" name="X83B113007DF2BB69"></a></p>

<h5>2.5-8 SpinSymClassFusion2ASin2SS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2ASin2SS</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.(Alt(k){\times}Sym(l))\)</span> and <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.(Alt(k){\times}Sym(l))\)</span> and <span class="SimpleMath">\(2.(Sym(k){\times}Sym(l))\)</span>, respectively, a corresponding class fusion map is determined analogously to <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.6)]</a>.</p>

<p><a id="X786B6E2D7CF13B51" name="X786B6E2D7CF13B51"></a></p>

<h5>2.5-9 SpinSymClassFusion2AAin2SA</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2AAin2SA</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.(Sym(k){\times}Alt(l))\)</span>, respectively, a corresponding class fusion map is determined. See <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.7)]</a>.</p>

<p><a id="X80DFBE347B87AA43" name="X80DFBE347B87AA43"></a></p>

<h5>2.5-10 SpinSymClassFusion2AAin2AS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2AAin2AS</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.(Alt(k){\times}Sym(l))\)</span>.</p>

<p>Given lists of class parameters <var class="Arg">cclSource</var> and <var class="Arg">cclDest</var> of (ordinary or <span class="SimpleMath">\(p\)</span>-modular) SpinSym tables of <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.(Alt(k){\times}Sym(l))\)</span>, respectively, a corresponding class fusion map is determined analogously to <a href="chapBib_mj.html#biBMaas2011">[2, (5.4.7)]</a>.</p>

<p><a id="X796897DF78143E05" name="X796897DF78143E05"></a></p>

<h5>2.5-11 SpinSymClassFusion2AAin2A</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpinSymClassFusion2AAin2A</code>( <var class="Arg">cclSource</var>, <var class="Arg">cclDest</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a fusion map between the SpinSym tables of <span class="SimpleMath">\(2.(Alt(k){\times}Alt(l))\)</span> and <span class="SimpleMath">\(2.Alt(k+l)\)</span>.</p>

--> --------------------

--> maximum size reached

--> --------------------

Messung V0.5
C=96 H=100 G=97

¤ Dauer der Verarbeitung: 0.29 Sekunden  ¤

*© 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 und die Messung sind noch experimentell.