# wedderga, chapter 7
#
# 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( "wedderga07.tst");
# doc/div-alg.xml:29-48
gap> G:=SmallGroup(48,15);
<pc group of size 48 with 5 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 5 generators>
gap> WedderburnDecompositionInfo(R);
[ [ 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> WedderburnDecompositionWithDivAlgParts(R);
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
[ 2, Rationals ], [ 2, Rationals ], [ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ],
[ 2, CF(3) ],
[ 2,
rec( Center := Rationals, DivAlg := true,
LocalIndices := [ [ 2, 2 ], [ 3, 2 ] ], SchurIndex := 2 ) ] ]
# doc/div-alg.xml:70-94
gap> G:=SmallGroup(240,89);
<permutation group of size 240 with 2 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 2 generators>
gap> W:=WedderburnDecompositionInfo(R);
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 ] ] ]
gap> CyclotomicAlgebraWithDivAlgPart(W[3]);
[ 2, rec( Center := Rationals, DivAlg := true,
LocalIndices := [ [ 5, 2 ], [ infinity, 2 ] ], SchurIndex := 2 ) ]
gap> CyclotomicAlgebraWithDivAlgPart(W[9]);
[ 2, rec( Center := NF(12,[ 1, 11 ]), DivAlg := true,
LocalIndices := [ [ infinity, 2 ] ], SchurIndex := 2 ) ]
gap> CyclotomicAlgebraWithDivAlgPart(W[10]);
[ 3, rec( Center := NF(8,[ 1, 7 ]), DivAlg := true,
LocalIndices := [ [ infinity, 2 ] ], SchurIndex := 2 ) ]
# doc/div-alg.xml:134-153
gap> G:=SmallGroup(63,1);
<pc group of size 63 with 3 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 3 generators>
gap> W:=WedderburnDecompositionInfo(R);
[ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(9) ], [ 3, NF(7,[ 1, 2, 4 ]) ],
[ 1, NF(21,[ 1, 4, 16 ]), 21, [ 3, 4, 7 ] ] ]
gap> SchurIndex(W[5]);
3
gap> G:=SmallGroup(40,3);
<pc group of size 40 with 4 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> SchurIndexByCharacter(GaussianRationals,G,Irr(G)[i]);
2
gap> SchurIndexByCharacter(CF(3),G,i);
1
# doc/div-alg.xml:194-219
gap> G:=SmallGroup(63,1);
<pc group of size 63 with 3 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 3 generators>
gap> W:=WedderburnDecompositionInfo(R);
[ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(9) ], [ 3, NF(7,[ 1, 2, 4 ]) ],
[ 1, NF(21,[ 1, 4, 16 ]), 21, [ 3, 4, 7 ] ] ]
gap> WedderburnDecompositionWithDivAlgParts(R);
[ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(9) ], [ 3, NF(7,[ 1, 2, 4 ]) ],
[ 1,
rec( Center := NF(21,[ 1, 4, 16 ]), DivAlg := true,
LocalIndices := [ [ 7, 3 ] ], SchurIndex := 3 ) ] ]
gap> WedderburnDecompositionAsSCAlgebras(R);
[ Rationals, CF(3), CF(9), ( NF(7,[ 1, 2, 4 ])^[ 3, 3 ] ),
<algebra of dimension 9 over NF(21,[ 1, 4, 16 ])> ]
gap> CyclotomicAlgebraAsSCAlgebra(W[5]);
<algebra of dimension 9 over NF(21,[ 1, 4, 16 ])>
gap> Size(Irr(G));
15
gap> SimpleComponentByCharacterAsSCAlgebra(Rationals,G,15);
<algebra of dimension 9 over NF(21,[ 1, 4, 16 ])>
# doc/div-alg.xml:243-250
gap> PPartOfN(2275,5);
25
gap> PDashPartOfN(2275,5);
91
# doc/div-alg.xml:267-277
gap> PSplitSubextension(Rationals,60,5);
GaussianRationals
gap> PSplitSubextension(NF(5,[1,4]),70,2);
NF(35,[ 1, 4, 9, 11, 16, 29 ])
gap> PSplitSubextension(NF(40,[1,9,11,19]),20,2);
NF(40,[ 1, 9, 11, 19])
# doc/div-alg.xml:300-320
gap> F:=CF(12);
CF(12)
gap> K:=NF(120,[1,49]) # Note that F is a subfield of K, with index 4.
> ; # Then we can find e, f, and g for the extension K/F at the prime 5.
NF(120,[ 1, 49 ])
gap> RamificationIndexAtP(F,120,5); RamificationIndexAtP(K,120,5); last2/last;
4
2
2
gap> ResidueDegreeAtP(F,120,5); ResidueDegreeAtP(K,120,5); last2/last;
1
1
1
gap> SplittingDegreeAtP(F,120,5); SplittingDegreeAtP(K,120,5); last2/last;
2
1
2
# doc/div-alg.xml:380-389
gap> A := [ 1, Rationals, 20, [ [ 2, 11, 0 ], [ 4, 3, 0 ] ], [ [ 0 ] ] ];;
gap> GlobalSplittingOfCyclotomicAlgebra(A);
[ 8, Rationals ]
gap> A := [ 1, Rationals, 20, [ [ 2, 11, 0 ], [ 4, 3, 10 ] ], [ [ 0 ] ] ];;
gap> GlobalSplittingOfCyclotomicAlgebra(A);
[ 2, Rationals, 10, [ 4, 3, 5 ] ]
# doc/div-alg.xml:438-448
gap> G:=PSU(3,3);
<permutation group of size 6048 with 2 generators>
gap> sc := SimpleComponentByCharacterDescent(Rationals,G,8);;
gap> sc{[1..3]}; # the 4th entry is [ 2, 5, 3 ] or [ 2, 5, 9 ]
[ 21/2, GaussianRationals, 12 ]
gap> SchurIndex(sc);
1
# doc/div-alg.xml:470-483
gap> G:=SmallGroup(63,1);
<pc group of size 63 with 3 generators>
gap> GaloisRepsOfCharacters(Rationals,G);
[ 1, 2, 4, 10, 12 ]
gap> GaloisRepsOfCharacters(CF(9),G);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13 ]
gap> GaloisRepsOfCharacters(NF(7,[1,2,4]),G);
[ 1, 2, 4, 10, 11, 12, 14 ]
gap> GaloisRepsOfCharacters(CF(63),G);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ]
# doc/div-alg.xml:511-524
gap> G:=GL(3,3);
GL(3,3)
gap> Size(G);
11232
gap> WedderburnDecompositionByCharacterDescent(Rationals,G);
[ [ 1, Rationals ], [ 1, Rationals ], [ 12, Rationals ], [ 12, Rationals ],
[ 13, Rationals ], [ 13, Rationals ], [ 16, NF(13,[ 1, 3, 9 ]) ],
[ 16, NF(13,[ 1, 3, 9 ]) ], [ 26, Rationals ], [ 26, Rationals ],
[ 26, NF(8,[ 1, 3 ]) ], [ 26, NF(8,[ 1, 3 ]) ], [ 27, Rationals ],
[ 27, Rationals ], [ 39, Rationals ], [ 39, Rationals ] ]
# doc/div-alg.xml:551-558
gap> A:=[1,Rationals,6,[2,5,3]];
[ 1, Rationals, 6, [ 2, 5, 3 ] ]
gap> LocalIndicesOfCyclicCyclotomicAlgebra(A);
[ [ 3, 2 ], [ infinity, 2 ] ]
# doc/div-alg.xml:596-611
gap> A:=[1,CF(4),20,[4,13,15]];
[ 1, GaussianRationals, 20, [ 4, 13, 15 ] ]
gap> LocalIndexAtOddP(A,5);
4
gap> A:=[1,NF(8,[1,7]),8,[2,7,4]];
[ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ]
gap> LocalIndexAtInfty(A);
2
gap> A:=[1,CF(7),28,[2,15,14]];
[ 1, CF(7), 28, [ 2, 15, 14 ] ]
gap> LocalIndexAtTwo(A);
2
# doc/div-alg.xml:661-674
gap> G:=SmallGroup(480,600);
<pc group of size 480 with 7 generators>
gap> W:=WedderburnDecompositionInfo(GroupRing(Rationals,G));;
gap> Size(W);
27
gap> W[27];
[ 1, NF(5,[ 1, 4 ]), 60, [ [ 2, 11, 0 ], [ 2, 19, 30 ], [ 2, 31, 30 ] ],
[ [ 0, 45 ], [ 15 ] ] ]
gap> LocalIndicesOfCyclotomicAlgebra(W[27]);
[ [ infinity, 2 ] ]
# doc/div-alg.xml:676-688
gap> G:=SmallGroup(160,82);
<pc group of size 160 with 6 generators>
gap> W:=WedderburnDecompositionInfo(GroupRing(Rationals,G));;
gap> Size(W);
14
gap> W[14];
[ 1, Rationals, 20, [ [ 2, 11, 0 ], [ 4, 3, 0 ] ], [ [ 5 ] ] ]
gap> LocalIndicesOfCyclotomicAlgebra(W[14]);
[ [ 2, 2 ], [ 5, 2 ] ]
# doc/div-alg.xml:702-709
gap> A:=[3,Rationals,12,[[2,5,3],[2,7,0]],[[3]]];
[ 3, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ]
gap> RootOfDimensionOfCyclotomicAlgebra(A);
12
# doc/div-alg.xml:739-757
gap> G:=SmallGroup(48,15);
<pc group of size 48 with 5 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 5 generators>
gap> W:=WedderburnDecompositionInfo(R);;
gap> A:=W[10];
[ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ]
gap> g:=DefiningGroupOfCyclotomicAlgebra(A);;
gap> IdSmallGroup(g);
[ 48, 15 ]
gap> n:=DefiningCharacterOfCyclotomicAlgebra(A);
12
gap> SimpleComponentOfGroupRingByCharacter(Rationals,G,n)
> ;#Note:this cyclotomic algebra is isomorphic to the other by a change of basis.
[ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ]
# doc/div-alg.xml:772-783
gap> G:=SmallGroup(48,16);
<pc group of size 48 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> LocalIndexAtInftyByCharacter(Rationals,G,i);
2
gap> LocalIndexAtInftyByCharacter(CF(3),G,Irr(G)[i]);
1
# doc/div-alg.xml:816-832
gap> G:=SmallGroup(72,21);
<pc group of size 72 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> D:=DefectGroupOfConjugacyClassAtP(G,i,3);
Group([ f4, f5 ])
gap> IsCyclic(last);
false
gap> D:=DefectGroupsOfPBlock(G,Irr(G)[i],3);
Group( [ f4, f5 ] )^G
gap> IsCyclic(Representative(D));
false
gap> DefectOfCharacterAtP(G,Irr(G)[i],3);
2
# doc/div-alg.xml:874-887
gap> G:=SmallGroup(80,28);
<pc group of size 80 with 5 generators>
gap> T:=CharacterTable(G);;
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> S:=T mod 5;
BrauerTable( <pc group of size 80 with 5 generators>, 5 )
gap> LocalIndexAtPByBrauerCharacter(Rationals,G,i,5);
2
gap> FinFieldExt(Rationals,G,5,i,9);
2
# doc/div-alg.xml:889-899
gap> G:=SmallGroup(72,20);
<pc group of size 72 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> LocalIndexAtPByBrauerCharacter(Rationals,G,Irr(G)[i],3);
[ 2, "DGnotCyclic" ]
gap> LocalIndexAtPByBrauerCharacter(Rationals,G,i,2);
1
# doc/div-alg.xml:943-955
gap> G:=SmallGroup(48,15);
<pc group of size 48 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> LocalIndexAtOddPByCharacter(Rationals,G,Irr(G)[i],3);
2
gap> LocalIndexAtTwoByCharacter(Rationals,G,Irr(G)[i]);
2
gap> LocalIndexAtTwoByCharacter(CF(3),G,Irr(G)[i]);
1
# doc/div-alg.xml:1010-1029
gap> LocalIndicesOfRationalSymbolAlgebra(-1,-1);
[ [ infinity, 2 ], [ 2, 2 ] ]
gap> LocalIndicesOfRationalSymbolAlgebra(3,-1);
[ [ 2, 2 ], [ 3, 2 ] ]
gap> LocalIndicesOfRationalSymbolAlgebra(-3,2);
fail
gap> LocalIndicesOfRationalSymbolAlgebra(3,7);
[ [ 2, 2 ], [ 7, 2 ] ]
gap> A:=QuaternionAlgebra(Rationals,-30,-15);
<algebra-with-one of dimension 4 over Rationals>
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
[ [ 5, 2 ], [ infinity, 2 ] ]
gap> A:=QuaternionAlgebra(CF(5),3,-2);
<algebra-with-one of dimension 4 over CF(5)>
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
fail
# doc/div-alg.xml:1055-1070
gap> A:=QuaternionAlgebra(Rationals,-30,-15);
<algebra-with-one of dimension 4 over Rationals>
gap> IsRationalQuaternionAlgebraADivisionRing(A);
true
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
[ [ 5, 2 ], [ infinity, 2 ] ]
gap> A:=QuaternionAlgebra(Rationals,3,-2);
<algebra-with-one of dimension 4 over Rationals>
gap> IsRationalQuaternionAlgebraADivisionRing(A);
false
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
[ ]
# doc/div-alg.xml:1121-1134
gap> G:=SmallGroup(96,35);
<pc group of size 96 with 6 generators>
gap> W:=WedderburnDecompositionInfo(GroupRing(Rationals,G));;
gap> Size(W);
12
gap> A:=W[12];
[ 1, NF(8,[ 1, 7 ]), 24, [ [ 2, 7, 12 ], [ 2, 17, 9 ] ], [ [ 3 ] ] ]
gap> DecomposeCyclotomicAlgebra(A);
[ [ NF(8,[ 1, 7 ]), CF(8), [ -1 ] ],
[ NF(8,[ 1, 7 ]), NF(24,[ 1, 7 ]), [ E(8)+2*E(8)^2+E(8)^3 ] ] ]
# doc/div-alg.xml:1158-1176
gap> A:=[NF(24,[1,11]),CF(24),[-1]];
[ NF(24,[ 1, 11 ]), CF(24), [ -1 ] ]
gap> ConvertCyclicAlgToCyclicCyclotomicAlg(A);
[ 1, NF(24,[ 1, 11 ]), 24, [ 2, 11, 12 ] ]
gap> LocalIndicesOfCyclicCyclotomicAlgebra(last);
[ ]
gap> ConvertQuadraticAlgToQuaternionAlg(A);
<algebra-with-one of dimension 4 over NF(24,[ 1, 11 ])>
gap> b:=Basis(last);; b[1]^2; b[2]^2; b[3]^2; b[4]^2;
e
(-1)*e
(-1)*e
(-1)*e
gap> b[2]*b[3]+b[3]*b[2];
0*e
# doc/div-alg.xml:1203-1223
gap> A:=QuaternionAlgebra(CF(5),-3,-1);
<algebra-with-one of dimension 4 over CF(5)>
gap> ConvertQuaternionAlgToQuadraticAlg(A);
[ CF(5), CF(15), [ -1 ] ]
gap> ConvertCyclicAlgToCyclicCyclotomicAlg(last);
[ 1, CF(5), 30, [ 2, 11, 15 ] ]
gap> SchurIndex(last);
1
gap> ConvertCyclicCyclotomicAlgToCyclicAlg(last2);
[ 1, [ CF(5), CF(15), [ -1 ] ] ]
gap> ConvertQuadraticAlgToQuaternionAlg(last[2]);
<algebra-with-one of dimension 4 over CF(5)>
gap> b:=Basis(last);; b[1]^2; b[2]^2; b[3]^2; b[4]^2;
e
(-3)*e
(-1)*e
(-3)*e
gap> STOP_TEST("wedderga07.tst", 1 );
