Quelle chap4.html Sprache: HTML

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">Goto Chapter: <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>

<a href="chap4_mj.html">[MathJax on]</a>
<a id="X7C651C9C78398FFF" name="X7C651C9C78398FFF"></a>
<div class="ChapSects"><a href="chap4.html#X7C651C9C78398FFF">4 Idempotents</a>
<div class="ContSect"> <a href="chap4.html#X7DF49142844C278D">4.1 Computing idempotents from character table</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7BBEB4A084DBF0D6">4.1-1 PrimitiveCentralIdempotentsByCharacterTable</a>
</div></div>
<div class="ContSect"> <a href="chap4.html#X83F7CF1E87D02581">4.2 Testing lists of idempotents for completeness</a>

<div class="ContSSBlock">
 <a href="chap4.html#X81FCD27E812078F0">4.2-1 IsCompleteSetOfOrthogonalIdempotents</a>
</div></div>
<div class="ContSect"> <a href="chap4.html#X7C66102485AF5F80">4.3 Idempotents from Shoda pairs</a>

<div class="ContSSBlock">
 <a href="chap4.html#X78D597207D3030EA">4.3-1 PrimitiveCentralIdempotentsByESSP</a>
 <a href="chap4.html#X7B48EE1A7ECAB151">4.3-2 PrimitiveCentralIdempotentsByStrongSP</a>
 <a href="chap4.html#X82460B1285A0A7D7">4.3-3 PrimitiveCentralIdempotentsBySP</a>
</div></div>
<div class="ContSect"> <a href="chap4.html#X8577F9547FC58C4C">4.4 Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7E95CDF17C4D54DB">4.4-1 PrimitiveIdempotentsNilpotent</a>
 <a href="chap4.html#X8784570980B9B750">4.4-2 PrimitiveIdempotentsTrivialTwisting</a>
</div></div>
</div>

<h3>4 Idempotents</h3>

<a id="X7DF49142844C278D" name="X7DF49142844C278D"></a>

<h4>4.1 Computing idempotents from character table</h4>

<a id="X7BBEB4A084DBF0D6" name="X7BBEB4A084DBF0D6"></a>

<h5>4.1-1 PrimitiveCentralIdempotentsByCharacterTable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveCentralIdempotentsByCharacterTable</code>( <var class="Arg">FG</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A list of group algebra elements.

The input <var class="Arg">FG</var> should be a semisimple group algebra.

Returns the list of primitive central idempotents of <var class="Arg">FG</var> using the character table of G (<a href="chap9.html#X87B6505C7C2EE054">9.4</a>).

<div class="example"><pre>

gap> QS3 := GroupRing( Rationals, SymmetricGroup(3) );; 
gap> PrimitiveCentralIdempotentsByCharacterTable( QS3 );
[ (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) ]
gap> QG:=GroupRing( Rationals , SmallGroup(24,3) );
<algebra-with-one over Rationals, with 4 generators>
gap> FG:=GroupRing( CF(3) , SmallGroup(24,3) );
<algebra-with-one over CF(3), with 4 generators>
gap> pciQG := PrimitiveCentralIdempotentsByCharacterTable(QG);;
gap> pciFG := PrimitiveCentralIdempotentsByCharacterTable(FG);;
gap> Length(pciQG);
5
gap> Length(pciFG);
7

</pre></div>

<a id="X83F7CF1E87D02581" name="X83F7CF1E87D02581"></a>

<h4>4.2 Testing lists of idempotents for completeness</h4>

<a id="X81FCD27E812078F0" name="X81FCD27E812078F0"></a>

<h5>4.2-1 IsCompleteSetOfOrthogonalIdempotents</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCompleteSetOfOrthogonalIdempotents</code>( <var class="Arg">R</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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>.

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:

⋅ The sum of the elements of <var class="Arg">list</var> is the identity of <var class="Arg">R</var>,

⋅ e^2=e, for every e in <var class="Arg">list</var> and

⋅ e*f=0, if e and f are elements in different positions of <var class="Arg">list</var>.

No claim is made on the idempotents being central or primitive.

Note that the if a non-zero element t 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.

<div class="example"><pre>

gap> QS5 := GroupRing( Rationals, SymmetricGroup(5) );;
gap> idemp := PrimitiveCentralIdempotentsByCharacterTable( QS5 );;
gap> IsCompleteSetOfOrthogonalIdempotents( QS5, idemp );
true
gap> IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ) ] );
true
gap> IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ), One( QS5 ) ] );
false

</pre></div>

<a id="X7C66102485AF5F80" name="X7C66102485AF5F80"></a>

<h4>4.3 Idempotents from Shoda pairs</h4>

<a id="X78D597207D3030EA" name="X78D597207D3030EA"></a>

<h5>4.3-1 PrimitiveCentralIdempotentsByESSP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveCentralIdempotentsByESSP</code>( <var class="Arg">QG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A list of group algebra elements.

The input <var class="Arg">QG</var> should be a semisimple rational group algebra of a finite group G.

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">9.16</a>) of G.

If the list of primitive central idempotents given by the output is not complete (i.e. if the group G is not normally monomial (<a href="chap9.html#X7C8D47C180E0ACAD">9.18</a>)) then a warning is displayed.

<div class="example"><pre>

gap> QG:=GroupRing( Rationals, DihedralGroup(16) );; 
gap> PrimitiveCentralIdempotentsByESSP( QG );
[ (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 ]
gap> QG := GroupRing( Rationals, SmallGroup(24,12) );;
gap> PrimitiveCentralIdempotentsByESSP( QG );
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>

<a id="X7B48EE1A7ECAB151" name="X7B48EE1A7ECAB151"></a>

<h5>4.3-2 PrimitiveCentralIdempotentsByStrongSP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveCentralIdempotentsByStrongSP</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A list of group algebra elements.

The input <var class="Arg">FG</var> should be a semisimple group algebra of a finite group G whose coefficient field F is either a finite field or the field ℚ of rationals.

If F = ℚ 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">9.15</a>) of G.

If F 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 (K,H) of G and q-cyclotomic classes modulo the index of H in K (<a href="chap9.html#X800D8C5087D79DC8">9.19</a>).

If the list of primitive central idempotents given by the output is not complete (i.e. if the group G is not strongly monomial (<a href="chap9.html#X84C694978557EFE5">9.17</a>)) then a warning is displayed.

<div class="example"><pre>

gap> QG:=GroupRing( Rationals, AlternatingGroup(4) );; 
gap> PrimitiveCentralIdempotentsByStrongSP( QG );
[ (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) ]
gap> QG := GroupRing( Rationals, SmallGroup(24,3) );;
gap> PrimitiveCentralIdempotentsByStrongSP( QG );;
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
gap> FG := GroupRing( GF(2), Group((1,2,3)) );;
gap> PrimitiveCentralIdempotentsByStrongSP( FG );
[ (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) ]
gap> FG := GroupRing( GF(5), SmallGroup(24,3) );; 
gap> PrimitiveCentralIdempotentsByStrongSP( FG );;
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!

</pre></div>

<a id="X82460B1285A0A7D7" name="X82460B1285A0A7D7"></a>

<h5>4.3-3 PrimitiveCentralIdempotentsBySP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveCentralIdempotentsBySP</code>( <var class="Arg">QG</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: A list of group algebra elements.

The input should be a rational group algebra of a finite group G.

Returns a list containing all the primitive central idempotents e of the rational group algebra <var class="Arg">QG</var> such that χ(e)ne 0 for some irreducible monomial character χ of G.

The output is the list of all primitive central idempotents of <var class="Arg">QG</var> if and only if G is monomial, otherwise a warning message is displayed.

<div class="example"><pre>

gap> QG := GroupRing( Rationals, SymmetricGroup(4) );
<algebra-with-one over Rationals, with 2 generators>
gap> pci:=PrimitiveCentralIdempotentsBySP( QG );
[ (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) ]
gap> IsCompleteSetOfPCIs(QG,pci);
true
gap> QS5 := GroupRing( Rationals, SymmetricGroup(5) );;
gap> pci:=PrimitiveCentralIdempotentsBySP( QS5 );;
Wedderga: Warning!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
gap> IsCompleteSetOfPCIs( QS5 , pci );
false

</pre></div>

The output of <code class="func">PrimitiveCentralIdempotentsBySP</code> contains the output of <code class="func">PrimitiveCentralIdempotentsByStrongSP</code> (<a href="chap4.html#X7B48EE1A7ECAB151">4.3-2</a>), possibly properly.

<div class="example"><pre>

gap> QG := GroupRing( Rationals, SmallGroup(48,28) );;
gap> pci:=PrimitiveCentralIdempotentsBySP( QG );;
Wedderga: Warning!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
gap> Length(pci); 
6
gap> spci:=PrimitiveCentralIdempotentsByStrongSP( QG );; 
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
gap> Length(spci);
5
gap> IsSubset(pci,spci); 
true
gap> QG:=GroupRing(Rationals,SmallGroup(1000,86));
<algebra-with-one over Rationals, with 6 generators>
gap> IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsBySP(QG) );
true
gap> IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsByStrongSP(QG) );
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
false

</pre></div>

<a id="X8577F9547FC58C4C" name="X8577F9547FC58C4C"></a>

<h4>4.4 Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes</h4>

<a id="X7E95CDF17C4D54DB" name="X7E95CDF17C4D54DB"></a>

<h5>4.4-1 PrimitiveIdempotentsNilpotent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
Returns: A list of orthogonal primitive idempotents.

The input <var class="Arg">FG</var> should be a semisimple group algebra of a finite nilpotent group G whose coefficient field F is a finite field. <var class="Arg">H</var> and <var class="Arg">K</var> should form a strong Shoda pair (H,K) of G. <var class="Arg">args</var> is a list containing an epimorphism map <var class="Arg">epi</var> from N_G(K) to N_G(K)/K and a generator <var class="Arg">gq</var> of H/K. C is the |F|-cyclotomic class modulo [H:K] (w.r.t. the generator gq of H/K)

The output is a complete set of orthogonal primitive idempotents of the simple algebra FGe_C(G,H,K) (<a href="chap9.html#X8472ACCF802EC188">9.22</a>).

<div class="example"><pre>

gap> G:=DihedralGroup(8);; 
gap> F:=GF(3);; 
gap> FG:=GroupRing(F,G);;
gap> H:=StrongShodaPairs(G)[5][1];
Group([ f1*f2*f3, f3 ])
gap> K:=StrongShodaPairs(G)[5][2];
Group([ f1*f2 ])
gap> N:=Normalizer(G,K); 
Group([ f1*f2*f3, f3 ])
gap> epi:=NaturalHomomorphismByNormalSubgroup(N,K);
[ f1*f2*f3, f3 ] -> [ f1, f1 ]
gap> QHK:=Image(epi,H); 
Group([ f1, f1 ])
gap> gq:=MinimalGeneratingSet(QHK)[1]; 
f1
gap> C:=CyclotomicClasses(Size(F),Index(H,K))[2];
[ 1 ]
gap> PrimitiveIdempotentsNilpotent(FG,H,K,C,[epi,gq]);
[ (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>

<a id="X8784570980B9B750" name="X8784570980B9B750"></a>

<h5>4.4-2 PrimitiveIdempotentsTrivialTwisting</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
Returns: A list of orthogonal primitive idempotents.

The input <var class="Arg">FG</var> should be a semisimple group algebra of a finite group G whose coefficient field F is a finite field. <var class="Arg">H</var> and <var class="Arg">K</var> should form a strong Shoda pair (H,K) of G. <var class="Arg">args</var> is a list containing an epimorphism map <var class="Arg">epi</var> from N_G(K) to N_G(K)/K and a generator <var class="Arg">gq</var> of H/K. C is the |F|-cyclotomic class modulo [H:K] (w.r.t. the generator gq of H/K). The input parameters should be such that the simple component FGe_C(G,H,K) has a trivial twisting.

The output is a complete set of orthogonal primitive idempotents of the simple algebra FGe_C(G,H,K) (<a href="chap9.html#X8472ACCF802EC188">9.22</a>).

<div class="example"><pre>

gap> G:=DihedralGroup(8);; 
gap> F:=GF(3);; 
gap> FG:=GroupRing(F,G);;
gap> H:=StrongShodaPairs(G)[5][1];
Group([ f1*f2*f3, f3 ])
gap> K:=StrongShodaPairs(G)[5][2];
Group([ f1*f2 ])
gap> N:=Normalizer(G,K); 
Group([ f1*f2*f3, f3 ])
gap> epi:=NaturalHomomorphismByNormalSubgroup(N,K);
[ f1*f2*f3, f3 ] -> [ f1, f1 ]
gap> QHK:=Image(epi,H); 
Group([ f1, f1 ])
gap> gq:=MinimalGeneratingSet(QHK)[1]; 
f1
gap> C:=CyclotomicClasses(Size(F),Index(H,K))[2];
[ 1 ]
gap> PrimitiveIdempotentsTrivialTwisting(FG,H,K,C,[epi,gq]);
[ (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">Goto Chapter: <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 />
generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a>
</body>
</html>

quality95%

¤ Dauer der Verarbeitung: 0.19 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

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

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.