> 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])); trueFinally, 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 ]
> 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
> C:= CompanionMatrix(CyclotomicPolynomial(5)); > ok, h:= IsGLZConjugate(C, C^2); ok; true > C^2 eq h^-1 * C * h; trueWe 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[Next][Prev] [Right] [Left] [Up] [Index] [Root]