Conjugacy

ZClasses(G) : GrpMat -> SeqEnum, SeqEnum

    Homogeneously: BoolElt              Default: false

Given a finite integral or rational matrix group G, its GL_n(Q)-conjugacy class splits into finitely many GL(n, Z)-conjugacy classes. Representatives of these classes are constructed as the action of G on some G-invariant sublattices. More precisely, the GL(n, Z)-conjugacy classes are in bijection with the orbits of G-invariant lattices under the normalizer N of G in GL(n, Q).

A G-lattice L' belongs to a G-lattice L if L = ∑_i L' e_i where e₁, ..., e_r denote the central idempotents of the endomorphism ring of G. Further, L is called homogeneously decomposable if L belongs to itself.

The algorithm will first compute representatives L₁, ..., L_k of the orbits of homogeneously decomposable G-lattices under the action of N.

In a second step, it will then compute the G-lattices L_{i, j} belonging to L_i up to the action of N.

The second return value will then consist of a sequence of k sequences T₁, ..., T_k. The first element T_i[1] is the basis matrix of L_i, the following entries are basis matrices of the lattices L_{i, j}.

The first return value is a sequence of integral matrix groups describing the action of G on the lattices L_{1, 1}, L_{1, 2}, ... . Hence these groups correspond to the GL_n(Z)-conjugacy classes of G.

If Homogeneously is set to true, the function will only compute the homogeneously decomposable lattices L₁, ..., L_k and the corresponding matrix groups. (If G is reducible, this option is much faster, but will not yield all conjugacy classes / orbits of lattices.)

IsGLZConjugate(G, H) : GrpMat[RngInt], GrpMat[RngInt] -> BoolElt, GrpMatElt

Tests whether the finite integral matrix groups G and H are conjugate in GL_n(Z). If so, a matrix x such that G^x = H is also returned.

IsBravaisEquivalent(G, H) : GrpMat[RngInt], GrpMat[RngInt] -> BoolElt, GrpMatElt

Given two finite integral matrix groups G and H, tests whether their Bravais groups B(G) and B(H) are conjugate in GL_n(Z). If so, a matrix x such that B(G)^x = B(H) is also returned.

Note that this function does not need to compute the Bravais groups and hence it is faster than calling IsGLZConjugate on the Bravais groups directly.

If G and H are known to be Bravais groups, this function is usually more efficient than calling IsGLZConjugate.

IsGLQConjugate(G, H) : GrpMat, GrpMat -> BoolElt, GrpMatElt

    Al: MonStgElt                       Default: 

Tests whether the finite rational matrix groups G and H are conjugate in GL_n(Q). If so, a matrix x such that G^x = H is also returned.

There are currently two algorithms available. If the optional parameter Al equals "Aut", Magma will use the GModule-machinery together with the outer automorphism group of H. If Al is set to "ZClasses", Magma splits the GL(n, Q)-conjugacy class of H into GL_n(Z)-conjugacy classes and then decides whether an integral copy of G lies in one of these classes by several calls to IsGLZConjugate.

If Al is not provided, a sensible choice is made by the system.