| products/Sources/formale Sprachen/GAP/pkg/wedderga/doc/chap4.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 (Wedderga) - Chapter 4: Idempotents</ title>
< meta http-equiv= "content-type" content= "text/html; charset=UTF-8" />
< meta name= "generator" content= "GAPDoc2HTML" />
< link rel= "stylesheet" type= "text/css" href= "manual.css" />
< script src= "manual.js" type= "text/javascript"></ script>
< script type= "text/javascript">overwriteStyle();</ script>
</ head>
< body class= "chap4" 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="chap8.html">8</a> <a href="chap9.html">9</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="chap0.html#contents">[Contents]</a> <a href="chap3.html">[Previous Chapter]</a> <a href="chap5.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap4_mj.html">[MathJax on]</a></p>
<p><a id="X7C651C9C78398FFF" name="X7C651C9C78398FFF"></a></p>
<div class="ChapSects"><a href="chap4.html#X7C651C9C78398FFF">4 <span class="Heading">Idempotents</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DF49142844C278D">4.1 <span class="Heading">Computing idempotents from character table</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BBEB4A084DBF0D6">4.1-1 PrimitiveCentralIdempotentsByCharacterTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X83F7CF1E87D02581">4.2 <span class="Heading">Testing lists of idempotents for completeness</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81FCD27E812078F0">4.2-1 IsCompleteSetOfOrthogonalIdempotents</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7C66102485AF5F80">4.3 <span class="Heading">Idempotents from Shoda pairs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78D597207D3030EA">4.3-1 PrimitiveCentralIdempotentsByESSP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B48EE1A7ECAB151">4.3-2 PrimitiveCentralIdempotentsByStrongSP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82460B1285A0A7D7">4.3-3 PrimitiveCentralIdempotentsBySP</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8577F9547FC58C4C">4.4 <span class="Heading">Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E95CDF17C4D54DB">4.4-1 PrimitiveIdempotentsNilpotent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8784570980B9B750">4.4-2 PrimitiveIdempotentsTrivialTwisting</a></span>
</div></div>
</div>
<h3>4 <span class="Heading">Idempotents</span></h3>
<p><a id="X7DF49142844C278D" name="X7DF49142844C278D"></a></p>
<h4>4.1 <span class="Heading">Computing idempotents from character table</span></h4>
<p><a id="X7BBEB4A084DBF0D6" name="X7BBEB4A084DBF0D6"></a></p>
<h5>4.1-1 PrimitiveCentralIdempotentsByCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimitiveCentralIdempotentsByCharacterTable</code>( <var class="Arg">FG</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A list of group algebra elements.</p>
<p>The input <var class="Arg">FG</var> should be a semisimple group algebra.</p>
<p>Returns the list of primitive central idempotents of <var class="Arg">FG</var> using the character table of <span class="SimpleMath">G</span> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QS3 := GroupRing( Rationals, SymmetricGroup(3) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveCentralIdempotentsByCharacterTable( QS3 );</span>
[ (1/6)*()+(-1/6)*(2,3)+(-1/6)*(1,2)+(1/6)*(1,2,3)+(1/6)*(1,3,2)+(-1/6)*(1,3),
(2/3)*()+(-1/3)*(1,2,3)+(-1/3)*(1,3,2), (1/6)*()+(1/6)*(2,3)+(1/6)*(1,2)+(1/
6)*(1,2,3)+(1/6)*(1,3,2)+(1/6)*(1,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing( Rationals , SmallGroup(24,3) );</span>
<algebra-with-one over Rationals, with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing( CF(3) , SmallGroup(24,3) );</span>
<algebra-with-one over CF(3), with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">pciQG := PrimitiveCentralIdempotentsByCharacterTable(QG);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pciFG := PrimitiveCentralIdempotentsByCharacterTable(FG);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(pciQG);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(pciFG);</span>
7
</pre></div>
<p><a id="X83F7CF1E87D02581" name="X83F7CF1E87D02581"></a></p>
<h4>4.2 <span class="Heading">Testing lists of idempotents for completeness</span></h4>
<p><a id="X81FCD27E812078F0" name="X81FCD27E812078F0"></a></p>
<h5>4.2-1 IsCompleteSetOfOrthogonalIdempotents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsCompleteSetOfOrthogonalIdempotents</code>( <var class="Arg">R</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The input should be formed by a unital ring <var class="Arg">R</var> and a list <var class="Arg">list</var> of elements of <var class="Arg">R</var>.</p>
<p>Returns <code class="keyw">true</code> if the list <var class="Arg">list</var> is a complete list of orthogonal idempotents of <var class="Arg">R</var>. That is, the output is <code class="keyw">true</code> provided the following conditions are satisfied:</p>
<p><span class="SimpleMath">⋅</span> The sum of the elements of <var class="Arg">list</var> is the identity of <var class="Arg">R</var>,</p>
<p><span class="SimpleMath">⋅</span> <span class="SimpleMath">e^2=e</span>, for every <span class="SimpleMath">e</span> in <var class="Arg">list</var> and</p>
<p><span class="SimpleMath">⋅</span> <span class="SimpleMath">e*f=0</span>, if <span class="SimpleMath">e</span> and <span class="SimpleMath">f</span> are elements in different positions of <var class="Arg">list</var>.</p>
<p>No claim is made on the idempotents being central or primitive.</p>
<p>Note that the if a non-zero element <span class="SimpleMath">t</span> of <var class="Arg">R</var> appears in two different positions of <var class="Arg">list</var> then the output is <code class="keyw">false</code>, and that the list <var class="Arg">list</var> must not contain zeroes.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QS5 := GroupRing( Rationals, SymmetricGroup(5) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">idemp := PrimitiveCentralIdempotentsByCharacterTable( QS5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCompleteSetOfOrthogonalIdempotents( QS5, idemp );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ) ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ), One( QS5 ) ] );</span>
false
</pre></div>
<p><a id="X7C66102485AF5F80" name="X7C66102485AF5F80"></a></p>
<h4>4.3 <span class="Heading">Idempotents from Shoda pairs</span></h4>
<p><a id="X78D597207D3030EA" name="X78D597207D3030EA"></a></p>
<h5>4.3-1 PrimitiveCentralIdempotentsByESSP</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimitiveCentralIdempotentsByESSP</code>( <var class="Arg">QG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A list of group algebra elements.</p>
<p>The input <var class="Arg">QG</var> should be a semisimple rational group algebra of a finite group <span class="SimpleMath">G</span>.</p>
<p>The output is the list of primitive central idempotents of the group algebra <var class="Arg">QG</var> realizable by extremely strong Shoda pairs (<a href="chap9.html#X81B5CE0378DC4913"><span class="RefLink">9.16</span></a>) of <span class="SimpleMath">G</span>.</p>
<p>If the list of primitive central idempotents given by the output is not complete (i.e. if the group <span class="SimpleMath">G</span> is not <em> normally monomial</em> (<a href="chap9.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>)) then a warning is displayed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing( Rationals, DihedralGroup(16) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveCentralIdempotentsByESSP( QG );</span>
[ (1/16)*<identity> of ...+(1/16)*f1+(1/16)*f2+(1/16)*f3+(1/16)*f4+(1/
16)*f1*f2+(1/16)*f1*f3+(1/16)*f1*f4+(1/16)*f2*f3+(1/16)*f2*f4+(1/
16)*f3*f4+(1/16)*f1*f2*f3+(1/16)*f1*f2*f4+(1/16)*f1*f3*f4+(1/
16)*f2*f3*f4+(1/16)*f1*f2*f3*f4, (1/16)*<identity> of ...+(-1/16)*f1+(-1/
16)*f2+(1/16)*f3+(1/16)*f4+(1/16)*f1*f2+(-1/16)*f1*f3+(-1/16)*f1*f4+(-1/
16)*f2*f3+(-1/16)*f2*f4+(1/16)*f3*f4+(1/16)*f1*f2*f3+(1/16)*f1*f2*f4+(-1/
16)*f1*f3*f4+(-1/16)*f2*f3*f4+(1/16)*f1*f2*f3*f4,
(1/16)*<identity> of ...+(-1/16)*f1+(1/16)*f2+(1/16)*f3+(1/16)*f4+(-1/
16)*f1*f2+(-1/16)*f1*f3+(-1/16)*f1*f4+(1/16)*f2*f3+(1/16)*f2*f4+(1/
16)*f3*f4+(-1/16)*f1*f2*f3+(-1/16)*f1*f2*f4+(-1/16)*f1*f3*f4+(1/
16)*f2*f3*f4+(-1/16)*f1*f2*f3*f4, (1/16)*<identity> of ...+(1/16)*f1+(-1/
16)*f2+(1/16)*f3+(1/16)*f4+(-1/16)*f1*f2+(1/16)*f1*f3+(1/16)*f1*f4+(-1/
16)*f2*f3+(-1/16)*f2*f4+(1/16)*f3*f4+(-1/16)*f1*f2*f3+(-1/16)*f1*f2*f4+(1/
16)*f1*f3*f4+(-1/16)*f2*f3*f4+(-1/16)*f1*f2*f3*f4,
(1/4)*<identity> of ...+(-1/4)*f3+(1/4)*f4+(-1/4)*f3*f4,
(1/2)*<identity> of ...+(-1/2)*f4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">QG := GroupRing( Rationals, SmallGroup(24,12) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveCentralIdempotentsByESSP( QG );</span>
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
[ (1/24)*<identity> of ...+(1/24)*f1+(1/24)*f2+(1/24)*f3+(1/24)*f4+(1/
24)*f1*f2+(1/24)*f1*f3+(1/24)*f1*f4+(1/24)*f2^2+(1/24)*f2*f3+(1/
24)*f2*f4+(1/24)*f3*f4+(1/24)*f1*f2^2+(1/24)*f1*f2*f3+(1/24)*f1*f2*f4+(1/
24)*f1*f3*f4+(1/24)*f2^2*f3+(1/24)*f2^2*f4+(1/24)*f2*f3*f4+(1/24)*f1*f2^
2*f3+(1/24)*f1*f2^2*f4+(1/24)*f1*f2*f3*f4+(1/24)*f2^2*f3*f4+(1/24)*f1*f2^
2*f3*f4, (1/24)*<identity> of ...+(-1/24)*f1+(1/24)*f2+(1/24)*f3+(1/
24)*f4+(-1/24)*f1*f2+(-1/24)*f1*f3+(-1/24)*f1*f4+(1/24)*f2^2+(1/
24)*f2*f3+(1/24)*f2*f4+(1/24)*f3*f4+(-1/24)*f1*f2^2+(-1/24)*f1*f2*f3+(-1/
24)*f1*f2*f4+(-1/24)*f1*f3*f4+(1/24)*f2^2*f3+(1/24)*f2^2*f4+(1/
24)*f2*f3*f4+(-1/24)*f1*f2^2*f3+(-1/24)*f1*f2^2*f4+(-1/24)*f1*f2*f3*f4+(1/
24)*f2^2*f3*f4+(-1/24)*f1*f2^2*f3*f4, (1/6)*<identity> of ...+(-1/12)*f2+(
1/6)*f3+(1/6)*f4+(-1/12)*f2^2+(-1/12)*f2*f3+(-1/12)*f2*f4+(1/6)*f3*f4+(-1/
12)*f2^2*f3+(-1/12)*f2^2*f4+(-1/12)*f2*f3*f4+(-1/12)*f2^2*f3*f4 ]
</pre></div>
<p><a id="X7B48EE1A7ECAB151" name="X7B48EE1A7ECAB151"></a></p>
<h5>4.3-2 PrimitiveCentralIdempotentsByStrongSP</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimitiveCentralIdempotentsByStrongSP</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A list of group algebra elements.</p>
<p>The input <var class="Arg">FG</var> should be a semisimple group algebra of a finite group <span class="SimpleMath">G</span> whose coefficient field <span class="SimpleMath">F</span> is either a finite field or the field <span class="SimpleMath">ℚ</span> of rationals.</p>
<p>If <span class="SimpleMath">F = ℚ</span> then the output is the list of primitive central idempotents of the group algebra <var class="Arg">FG</var> realizable by strong Shoda pairs (<a href="chap9.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath">G</span>.</p>
<p>If <span class="SimpleMath">F</span> is a finite field then the output is the list of primitive central idempotents of <var class="Arg">FG</var> realizable by strong Shoda pairs <span class="SimpleMath">(K,H)</span> of <span class="SimpleMath">G</span> and <span class="SimpleMath">q</span>-cyclotomic classes modulo the index of <span class="SimpleMath">H</span> in <span class="SimpleMath">K</span> (<a href="chap9.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>).</p>
<p>If the list of primitive central idempotents given by the output is not complete (i.e. if the group <span class="SimpleMath">G</span> is not <em>strongly monomial</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>)) then a warning is displayed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing( Rationals, AlternatingGroup(4) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveCentralIdempotentsByStrongSP( QG );</span>
[ (1/12)*()+(1/12)*(2,3,4)+(1/12)*(2,4,3)+(1/12)*(1,2)(3,4)+(1/12)*(1,2,3)+(1/
12)*(1,2,4)+(1/12)*(1,3,2)+(1/12)*(1,3,4)+(1/12)*(1,3)(2,4)+(1/12)*
(1,4,2)+(1/12)*(1,4,3)+(1/12)*(1,4)(2,3),
(1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(-1/12)*(1,2,3)+(
-1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)(2,4)+(-1/12)*
(1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3),
(3/4)*()+(-1/4)*(1,2)(3,4)+(-1/4)*(1,3)(2,4)+(-1/4)*(1,4)(2,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">QG := GroupRing( Rationals, SmallGroup(24,3) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveCentralIdempotentsByStrongSP( QG );;</span>
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
<span class="GAPprompt">gap></span> <span class="GAPinput">FG := GroupRing( GF(2), Group((1,2,3)) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveCentralIdempotentsByStrongSP( FG );</span>
[ (Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2),
(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FG := GroupRing( GF(5), SmallGroup(24,3) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveCentralIdempotentsByStrongSP( FG );;</span>
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
</pre></div>
<p><a id="X82460B1285A0A7D7" name="X82460B1285A0A7D7"></a></p>
<h5>4.3-3 PrimitiveCentralIdempotentsBySP</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimitiveCentralIdempotentsBySP</code>( <var class="Arg">QG</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A list of group algebra elements.</p>
<p>The input should be a rational group algebra of a finite group <span class="SimpleMath">G</span>.</p>
<p>Returns a list containing all the primitive central idempotents <span class="SimpleMath">e</span> of the rational group algebra <var class="Arg">QG</var> such that <span class="SimpleMath">χ(e)ne 0</span> for some irreducible monomial character <span class="SimpleMath">χ</span> of <span class="SimpleMath">G</span>.</p>
<p>The output is the list of all primitive central idempotents of <var class="Arg">QG</var> if and only if <span class="SimpleMath">G</span> is monomial, otherwise a warning message is displayed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG := GroupRing( Rationals, SymmetricGroup(4) );</span>
<algebra-with-one over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">pci:=PrimitiveCentralIdempotentsBySP( QG );</span>
[ (1/24)*()+(1/24)*(3,4)+(1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*(2,4,3)+(1/24)*
(2,4)+(1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(1/24)*(1,2,3,4)+(1/
24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(1/24)*(1,3,4,2)+(1/24)*
(1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(1/24)*(1,3,2,4)+(1/24)*(1,4,3,2)+(
1/24)*(1,4,2)+(1/24)*(1,4,3)+(1/24)*(1,4)+(1/24)*(1,4,2,3)+(1/24)*(1,4)
(2,3), (1/24)*()+(-1/24)*(3,4)+(-1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*
(2,4,3)+(-1/24)*(2,4)+(-1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(-1/
24)*(1,2,3,4)+(-1/24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(-1/24)*
(1,3,4,2)+(-1/24)*(1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(-1/24)*
(1,3,2,4)+(-1/24)*(1,4,3,2)+(1/24)*(1,4,2)+(1/24)*(1,4,3)+(-1/24)*(1,4)+(
-1/24)*(1,4,2,3)+(1/24)*(1,4)(2,3), (3/8)*()+(-1/8)*(3,4)+(-1/8)*(2,3)+(
-1/8)*(2,4)+(-1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(1/8)*(1,2,3,4)+(1/8)*
(1,2,4,3)+(1/8)*(1,3,4,2)+(-1/8)*(1,3)+(-1/8)*(1,3)(2,4)+(1/8)*(1,3,2,4)+(
1/8)*(1,4,3,2)+(-1/8)*(1,4)+(1/8)*(1,4,2,3)+(-1/8)*(1,4)(2,3),
(3/8)*()+(1/8)*(3,4)+(1/8)*(2,3)+(1/8)*(2,4)+(1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(
-1/8)*(1,2,3,4)+(-1/8)*(1,2,4,3)+(-1/8)*(1,3,4,2)+(1/8)*(1,3)+(-1/8)*(1,3)
(2,4)+(-1/8)*(1,3,2,4)+(-1/8)*(1,4,3,2)+(1/8)*(1,4)+(-1/8)*(1,4,2,3)+(-1/
8)*(1,4)(2,3), (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(
-1/12)*(1,2,3)+(-1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)
(2,4)+(-1/12)*(1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCompleteSetOfPCIs(QG,pci);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">QS5 := GroupRing( Rationals, SymmetricGroup(5) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pci:=PrimitiveCentralIdempotentsBySP( QS5 );;</span>
Wedderga: Warning!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCompleteSetOfPCIs( QS5 , pci );</span>
false
</pre></div>
<p>The output of <code class="func">PrimitiveCentralIdempotentsBySP</code> contains the output of <code class="func">PrimitiveCentralIdempotentsByStrongSP</code> (<a href="chap4.html#X7B48EE1A7ECAB151"><span class="RefLink">4.3-2</span></a>), possibly properly.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG := GroupRing( Rationals, SmallGroup(48,28) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pci:=PrimitiveCentralIdempotentsBySP( QG );;</span>
Wedderga: Warning!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(pci); </span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">spci:=PrimitiveCentralIdempotentsByStrongSP( QG );; </span>
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(spci);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset(pci,spci); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing(Rationals,SmallGroup(1000,86));</span>
<algebra-with-one over Rationals, with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsBySP(QG) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsByStrongSP(QG) );</span>
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
false
</pre></div>
<p><a id="X8577F9547FC58C4C" name="X8577F9547FC58C4C"></a></p>
<h4>4.4 <span class="Heading">Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes</span></h4>
<p><a id="X7E95CDF17C4D54DB" name="X7E95CDF17C4D54DB"></a></p>
<h5>4.4-1 PrimitiveIdempotentsNilpotent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimitiveIdempotentsNilpotent</code>( <var class="Arg">FG</var>, <var class="Arg">H</var>, <var class="Arg">K</var>, <var class="Arg">C</var>, <var class="Arg">args</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A list of orthogonal primitive idempotents.</p>
<p>The input <var class="Arg">FG</var> should be a semisimple group algebra of a finite nilpotent group <span class="SimpleMath">G</span> whose coefficient field <span class="SimpleMath">F</span> is a finite field. <var class="Arg">H</var> and <var class="Arg">K</var> should form a strong Shoda pair <span class="SimpleMath">(H,K)</span> of <span class="SimpleMath">G</span>. <var class="Arg">args</var> is a list containing an epimorphism map <var class="Arg">epi</var> from <span class="SimpleMath">N_G(K)</span> to <span class="SimpleMath">N_G(K)/K</span> and a generator <var class="Arg">gq</var> of <span class="SimpleMath">H/K</span>. <span class="SimpleMath">C</span> is the <span class="SimpleMath">|F|</span>-cyclotomic class modulo <span class="SimpleMath">[H:K]</span> (w.r.t. the generator <span class="SimpleMath">gq</span> of <span class="SimpleMath">H/K</span>)</p>
<p>The output is a complete set of orthogonal primitive idempotents of the simple algebra <span class="SimpleMath">FGe_C(G,H,K)</span> (<a href="chap9.html#X8472ACCF802EC188"><span class="RefLink">9.22</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=DihedralGroup(8);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=GF(3);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing(F,G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=StrongShodaPairs(G)[5][1];</span>
Group([ f1*f2*f3, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=StrongShodaPairs(G)[5][2];</span>
Group([ f1*f2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=Normalizer(G,K); </span>
Group([ f1*f2*f3, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">epi:=NaturalHomomorphismByNormalSubgroup(N,K);</span>
[ f1*f2*f3, f3 ] -> [ f1, f1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">QHK:=Image(epi,H); </span>
Group([ f1, f1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">gq:=MinimalGeneratingSet(QHK)[1]; </span>
f1
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=CyclotomicClasses(Size(F),Index(H,K))[2];</span>
[ 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveIdempotentsNilpotent(FG,H,K,C,[epi,gq]);</span>
[ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3,
(Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ]
</pre></div>
<p><a id="X8784570980B9B750" name="X8784570980B9B750"></a></p>
<h5>4.4-2 PrimitiveIdempotentsTrivialTwisting</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimitiveIdempotentsTrivialTwisting</code>( <var class="Arg">FG</var>, <var class="Arg">H</var>, <var class="Arg">K</var>, <var class="Arg">C</var>, <var class="Arg">args</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A list of orthogonal primitive idempotents.</p>
<p>The input <var class="Arg">FG</var> should be a semisimple group algebra of a finite group <span class="SimpleMath">G</span> whose coefficient field <span class="SimpleMath">F</span> is a finite field. <var class="Arg">H</var> and <var class="Arg">K</var> should form a strong Shoda pair <span class="SimpleMath">(H,K)</span> of <span class="SimpleMath">G</span>. <var class="Arg">args</var> is a list containing an epimorphism map <var class="Arg">epi</var> from <span class="SimpleMath">N_G(K)</span> to <span class="SimpleMath">N_G(K)/K</span> and a generator <var class="Arg">gq</var> of <span class="SimpleMath">H/K</span>. <span class="SimpleMath">C</span> is the <span class="SimpleMath">|F|</span>-cyclotomic class modulo <span class="SimpleMath">[H:K]</span> (w.r.t. the generator <span class="SimpleMath">gq</span> of <span class="SimpleMath">H/K</span>). The input parameters should be such that the simple component <span class="SimpleMath">FGe_C(G,H,K)</span> has a trivial twisting.</p>
<p>The output is a complete set of orthogonal primitive idempotents of the simple algebra <span class="SimpleMath">FGe_C(G,H,K)</span> (<a href="chap9.html#X8472ACCF802EC188"><span class="RefLink">9.22</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=DihedralGroup(8);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=GF(3);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing(F,G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=StrongShodaPairs(G)[5][1];</span>
Group([ f1*f2*f3, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=StrongShodaPairs(G)[5][2];</span>
Group([ f1*f2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=Normalizer(G,K); </span>
Group([ f1*f2*f3, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">epi:=NaturalHomomorphismByNormalSubgroup(N,K);</span>
[ f1*f2*f3, f3 ] -> [ f1, f1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">QHK:=Image(epi,H); </span>
Group([ f1, f1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">gq:=MinimalGeneratingSet(QHK)[1]; </span>
f1
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=CyclotomicClasses(Size(F),Index(H,K))[2];</span>
[ 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveIdempotentsTrivialTwisting(FG,H,K,C,[epi,gq]);</span>
[ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3,
(Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ]
</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap3.html">[Previous Chapter]</a> <a href="chap5.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="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chapBib.html">Bib</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>
¤ Dauer der Verarbeitung: 0.4 Sekunden
(vorverarbeitet am 2026-11-04)
¤
*© Formatika GbR, Deutschland
|
|
