# wedderga, chapter 4
#
# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!
#
# This file has been autogenerated with GAP. It contains examples
# extracted from the documentation. Each example is preceded by the
# comment which points to the location of its source.
#
gap> START_TEST( "wedderga04.tst");
# doc/idempot.xml:23-41
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
# doc/idempot.xml:80-91
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
# doc/idempot.xml:124-162
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 ]
# doc/idempot.xml:196-220
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!
# doc/idempot.xml:243-277
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
# doc/idempot.xml:283-307
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
# doc/idempot.xml:338-361
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 ]
# doc/idempot.xml:388-411
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 ]
gap> STOP_TEST("wedderga04.tst", 1 );
