# wedderga, chapter 2
#
# 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( "wedderga02.tst");
# doc/decomp.xml:31-45
gap> WedderburnDecomposition( GroupRing( GF(5), DihedralGroup(16) ) );
[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ),
( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), ( GF(5^2)^[ 2, 2 ] ) ]
gap> WedderburnDecomposition( GroupRing( Rationals, DihedralGroup(16) ) );
[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),
<crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), CF(8) ) of a group of size 2> ]
gap> WedderburnDecomposition( GroupRing( CF(5), DihedralGroup(16) ) );
[ CF(5), CF(5), CF(5), CF(5), ( CF(5)^[ 2, 2 ] ),
<crossed product with center NF(40,[ 1, 31 ]) over AsField( NF(40,
[ 1, 31 ]), CF(40) ) of a group of size 2> ]
# doc/decomp.xml:77-96
gap> WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) );
[ Rationals, Rationals, Rationals, Rationals,
<crossed product with center Rationals over CF(3) of a group of size 2>,
<crossed product with center Rationals over GaussianRationals of a group of \
size 2>, <crossed product with center Rationals over CF(3) of a group of size
2>, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), CF(8) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ),
<crossed product with center Rationals over CF(12) of a group of size 4> ]
gap> WedderburnDecomposition( GroupRing( CF(3), SmallGroup(48,15) ) );
[ CF(3), CF(3), CF(3), CF(3), ( CF(3)^[ 2, 2 ] ),
<crossed product with center CF(3) over AsField( CF(3), CF(
12) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ),
<crossed product with center NF(24,[ 1, 7 ]) over AsField( NF(24,
[ 1, 7 ]), CF(24) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ),
( CF(3)^[ 2, 2 ] ), ( <crossed product with center CF(3) over AsField( CF(
3), CF(12) ) of a group of size 2>^[ 2, 2 ] ) ]
# doc/decomp.xml:108-124
gap> QG:=GroupRing(Rationals,SmallGroup(240,89));
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecomposition(QG);
Wedderga: Warning!!!
Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!!
[ Rationals, Rationals, <crossed product with center Rationals over CF(
5) of a group of size 4>, ( Rationals^[ 4, 4 ] ), ( Rationals^[ 4, 4 ] ),
( Rationals^[ 5, 5 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 6, 6 ] ),
<crossed product with center NF(12,[ 1, 11 ]) over AsField( NF(12,
[ 1, 11 ]), NF(60,[ 1, 11 ]) ) of a group of size 4>,
[ 3/2, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), NF(40,[ 1, 31 ]) ) of a group of size 4> ] ]
# doc/decomp.xml:193-202
gap> WedderburnDecompositionInfo( GroupRing( Rationals, DihedralGroup(16) ) );
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
[ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ] ]
gap> WedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) );
[ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ],
[ 2, NF(40,[ 1, 31 ]) ] ]
# doc/decomp.xml:220-239
gap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;;rel:=[a^8,a^4*b^2,b^-1*a*b*a];;
gap> Q16:=F/rel;; QQ16:=GroupRing( Rationals, Q16 );;
gap> QS4:=GroupRing( Rationals, SymmetricGroup(4) );;
gap> WedderburnDecomposition(QQ16);
[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),
<crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), CF(8) ) of a group of size 2> ]
gap> WedderburnDecomposition( QS4 );
[ Rationals, Rationals, <crossed product with center Rationals over CF(
3) of a group of size 2>, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ) ]
gap> WedderburnDecompositionInfo(QQ16);
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
[ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ] ]
gap> WedderburnDecompositionInfo(QS4);
[ [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ], [ 3, Rationals ],
[ 3, Rationals ] ]
# doc/decomp.xml:293-304
gap> WedderburnDecompositionInfo( GroupRing( Rationals, SmallGroup(48,15) ) );
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
[ 2, Rationals ], [ 2, Rationals ], [ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ],
[ 2, CF(3) ], [ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ] ]
gap> WedderburnDecompositionInfo( GroupRing( CF(3), SmallGroup(48,15) ) );
[ [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 2, CF(3) ],
[ 2, CF(3) ], [ 2, CF(3) ], [ 2, NF(24,[ 1, 7 ]) ], [ 2, CF(3) ],
[ 2, CF(3) ], [ 4, CF(3) ] ]
# doc/decomp.xml:317-330
gap> QG:=GroupRing(Rationals,SmallGroup(240,89));
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecompositionInfo(QG);
Wedderga: Warning!!!
Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!!
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 10, [ 4, 3, 5 ] ],
[ 4, Rationals ], [ 4, Rationals ], [ 5, Rationals ], [ 5, Rationals ],
[ 6, Rationals ], [ 1, NF(12,[ 1, 11 ]), 10, [ 4, 3, 5 ] ],
[ 3/2, NF(8,[ 1, 7 ]), 10, [ 4, 3, 5 ] ] ]
# doc/decomp.xml:387-409
gap> A5 := AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> chi := First(Irr( A5 ), chi -> Degree(chi) = 3);;
gap> SimpleAlgebraByCharacter( GroupRing( Rationals, A5 ), chi );
( NF(5,[ 1, 4 ])^[ 3, 3 ] )
gap> SimpleAlgebraByCharacter( GroupRing( GF(7), A5 ), chi );
( GF(7^2)^[ 3, 3 ] )
gap> G:=SmallGroup(128,100);
<pc group of size 128 with 7 generators>
gap> chi4:=Filtered(Irr(G),x->Degree(x)=4);;
gap> List(chi4,x->SimpleAlgebraByCharacter(GroupRing(Rationals,G),x));
[ ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ) ]
# doc/decomp.xml:442-453
gap> G:=SmallGroup(128,100);
<pc group of size 128 with 7 generators>
gap> QG:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 7 generators>
gap> chi4:=Filtered(Irr(G),x->Degree(x)=4);;
gap> List(chi4,x->SimpleAlgebraByCharacterInfo(QG,x));
[ [ 4, NF(8,[ 1, 3 ]) ], [ 4, NF(8,[ 1, 3 ]) ], [ 4, NF(8,[ 1, 3 ]) ],
[ 4, NF(8,[ 1, 3 ]) ] ]
# doc/decomp.xml:522-536
gap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;
gap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;
gap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);;
gap> QG:=GroupRing( Rationals, G );;
gap> FG:=GroupRing( GF(7), G );;
gap> SimpleAlgebraByStrongSP( QG, K, H );
<crossed product over CF(16) of a group of size 2>
gap> SimpleAlgebraByStrongSP( FG, K, H, [1,7] );
( GF(7)^[ 2, 2 ] )
gap> SimpleAlgebraByStrongSP( FG, K, H, 1 );
( GF(7)^[ 2, 2 ] )
# doc/decomp.xml:596-612
gap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;
gap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;
gap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);;
gap> QG:=GroupRing( Rationals, G );;
gap> FG:=GroupRing( GF(7), G );;
gap> SimpleAlgebraByStrongSP( QG, K, H );
<crossed product over CF(16) of a group of size 2>
gap> SimpleAlgebraByStrongSPInfo( QG, K, H );
[ 1, NF(16,[ 1, 7 ]), 16, [ [ 2, 7, 8 ] ], [ ] ]
gap> SimpleAlgebraByStrongSPInfo( FG, K, H, [1,7] );
[ 2, 7 ]
gap> SimpleAlgebraByStrongSPInfo( FG, K, H, 1 );
[ 2, 7 ]
gap> STOP_TEST("wedderga02.tst", 1 );
