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

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7BBEB4A084DBF0D6">4.1-1 PrimitiveCentralIdempotentsByCharacterTable</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X83F7CF1E87D02581">4.2 Testing lists of idempotents for completeness</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X81FCD27E812078F0">4.2-1 IsCompleteSetOfOrthogonalIdempotents</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X7C66102485AF5F80">4.3 Idempotents from Shoda pairs</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X78D597207D3030EA">4.3-1 PrimitiveCentralIdempotentsByESSP</a>
 <a href="chap4_mj.html#X7B48EE1A7ECAB151">4.3-2 PrimitiveCentralIdempotentsByStrongSP</a>
 <a href="chap4_mj.html#X82460B1285A0A7D7">4.3-3 PrimitiveCentralIdempotentsBySP</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X8577F9547FC58C4C">4.4 Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7E95CDF17C4D54DB">4.4-1 PrimitiveIdempotentsNilpotent</a>
 <a href="chap4_mj.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">&#an style='color: green'>8227; 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_mj.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">&#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>
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:

\(\cdot\) The sum of the elements of <var class="Arg">list</var> is the identity of <var class="Arg">R</var>,

\(\cdot\) \(e^2=e\), for every \(e\) in <var class="Arg">list</var> and

\(\cdot\) \(e*f=0\), 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">&#an style='color: green'>8227; 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_mj.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_mj.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">&#an style='color: green'>8227; 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_mj.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_mj.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_mj.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">&#an style='color: green'>8227; 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 \(\chi(e)\ne 0\) for some irreducible monomial character \(\chi\) 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_mj.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">&#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>
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_mj.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">&#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>
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_mj.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>

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