<?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" /> </head> <body> <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="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap3.html">Previous Chapter</a> <a href="chap5.html">Next Chapter</a> </div> <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="nocss"> </span><a href="chap4.html#X7DF49142844C278D">4.1 <span class="Heading">Computing idempotents from character table</span></a> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BBEB4A084DBF0D6">4.1-1 PrimitiveCentralIdempotentsByCharacterTable</a></span> </div> <div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X83F7CF1E87D02581">4.2 <span class="Heading">Testing lists of idempotents for completeness</span></a> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81FCD27E812078F0">4.2-1 IsCompleteSetOfOrthogonalIdempotents</a></span> </div> <div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7C66102485AF5F80">4.3 <span class="Heading">Idempotents from Shoda pairs</span></a> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B48EE1A7ECAB151">4.3-1 PrimitiveCentralIdempotentsByStrongSP</a></span> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82460B1285A0A7D7">4.3-2 PrimitiveCentralIdempotentsBySP</a></span> </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">> PrimitiveCentralIdempotentsByCharacterTable</code>( <var class="Arg">FG</var> )</td><td class="tdright">( operation )</td></tr></table></div> <p><b>Returns: </b>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 G (<a href="chap7.html#X87B6505C7C2EE054"><b>7.4</b></a>).</p> <table class="example"> <tr><td><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></td></tr></table> <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">> IsCompleteSetOfOrthogonalIdempotents</code>( <var class="Arg">R, 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>* The sum of the elements of <var class="Arg">list</var> is the identity of <var class="Arg">R</var>,</p> <p>* e^2=e, for every e in <var class="Arg">list</var> and</p> <p>* e*f=0, if e and f 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 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.</p> <table class="example"> <tr><td><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></td></tr></table> <p><a id="X7C66102485AF5F80" name="X7C66102485AF5F80"></a></p> <h4>4.3 <span class="Heading">Idempotents from Shoda pairs</span></h4> <p><a id="X7B48EE1A7ECAB151" name="X7B48EE1A7ECAB151"></a></p> <h5>4.3-1 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> <p><b>Returns: </b>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 G whose coefficient field F is either a finite field or the field ℚ of rationals.</p> <p>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="chap7.html#X81DAF5267D30C83A"><b>7.15</b></a>) of G.</p> <p>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="chap7.html#X800D8C5087D79DC8"><b>7.17</b></a>).</p> <p>If the list of primitive central idempotents given by the output is not complete (i.e. if the group G is not <em>strongly monomial</em> (<a href="chap7.html#X84C694978557EFE5"><b>7.16</b></a>)) then a warning is displayed.</p> <table class="example"> <tr><td><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></td></tr></table> <p><a id="X82460B1285A0A7D7" name="X82460B1285A0A7D7"></a></p> <h5>4.3-2 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> <p><b>Returns: </b>A list of group algebra elements.</p> <p>The input should be a rational group algebra of a finite group G.</p> <p>Returns a list containing all the primitive central idempotents e of the rational group algebra <var class="Arg">QG</var> such that chi(e)ne 0 for some irreducible monomial character chi of G.</p> <p>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.</p> <table class="example"> <tr><td><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></td></tr></table> <p>The output of <code class="func">PrimitiveCentralIdempotentsBySP</code> contains the output of <code class="func">PrimitiveCentralIdempotentsByStrongSP</code> (<a href="chap4.html#X7B48EE1A7ECAB151"><b>4.3-1</b></a>), possibly properly.</p> <table class="example"> <tr><td><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></td></tr></table> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</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="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>