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Examples


Example GrpMatQZ_ZClasses (H62E1)

We split the GL3(Q)-conjugacy class of the following faithful representation of the dihedral group with 12 elements.

> G := MatrixGroup< 3, Integers() |
>  [ 1, -1, 0, 0, -1, 0, 0, 0, 1 ],
>  [ 1, -1, 0, 1, 0, 0, 0, 0, -1 ] >;
> Z, T:= ZClasses(G);
> #Z;
3
> < #t : t in T >;
<1, 2>
So there are 2 classes of homogeneously decomposable lattices represented by T[1,1] and T[2,1]. The third lattice T[2,2] belongs to T[2,1] as we check.

> Q := Rationals();
> GQ := ChangeRing(G, Q);
> Ids := CentralIdempotents(EndomorphismRing(GQ));
> L := VerticalJoin([ Matrix(Integers(), T[2,2] * i) : i in Ids]);
> Image(L) eq Image(Matrix(Integers(), T[2,1]));
true
Finally, we check that the 3 GL3(Z)-conjugacy classes stored in Z correspond to the 3 lattices in T.

> TT := &cat T;
> [ GQ eq ChangeRing(Z[i], Q)^(GL(3, Q) ! TT[i]) : i in [1..#Z] ];
[ true, true, true ]

Example GrpMatQZ_conjugacy (H62E2)

We test that the automorphism groups of the lattices B8 and D8 are conjugate in GL8(Q) but not in GL8(Z).

> G := AutomorphismGroup( Lattice("B", 8) );
> H := AutomorphismGroup( Lattice("D", 8) );
> ok, x := IsGLQConjugate(G, H); ok, x;
true
[ 1 -1  0  0  0  0  0  0]
[ 1 -1 -2  0  0  0  0  0]
[-1  1  2  2  2  2  2  2]
[ 1  1  0  0  0  0  0  0]
[-1  1  2  2  2  2  2  0]
[ 1 -1 -2 -2 -2  0  0  0]
[-1  1  2  2  2  2  0  0]
[-1  1  2  2  0  0  0  0]
> Determinant(x);
-128
> IsGLZConjugate(G,H);
false

Example GrpMatQZ_conjugacy_matrices (H62E3)

Let C be the companion matrix of the fifth cyclotomic polyomial. We find a unimodular matrix that induces the automorphism C - > C2.

> C:= CompanionMatrix(CyclotomicPolynomial(5));
> ok, h:= IsGLZConjugate(C, C^2); ok;
true
> C^2 eq h^-1 * C * h;
true
We now check by hand that this automorphism cannot be realized by a matrix of determiant 1.

> Determinant(h);
-1
> G:= CentralizerGLZ(C);
> [ Determinant(g) : g in Generators(G) ];
[1, 1, 1]
Of course, we could also just ask:

> IsSLZConjugate(C, C^2);
false
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013