Homogeneously: BoolElt Default: false
Given a finite integral or rational matrix group G, its GLn(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' ei where e1, ..., er 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, ..., Lk of the orbits of homogeneously decomposable G-lattices under the action of N.
In a second step, it will then compute the G-lattices Li, j belonging to Li up to the action of N.
The second return value will then consist of a sequence of k sequences T1, ..., Tk. The first element Ti[1] is the basis matrix of Li, the following entries are basis matrices of the lattices Li, j.
The first return value is a sequence of integral matrix groups describing the action of G on the lattices L1, 1, L1, 2, ... . Hence these groups correspond to the GLn(Z)-conjugacy classes of G.
If Homogeneously is set to true, the function will only compute the homogeneously decomposable lattices L1, ..., Lk and the corresponding matrix groups. (If G is reducible, this option is much faster, but will not yield all conjugacy classes / orbits of lattices.)
Tests whether the finite integral matrix groups G and H are conjugate in GLn(Z). If so, a matrix x such that Gx = H is also returned.
Given two finite integral matrix groups G and H, tests whether their Bravais groups B(G) and B(H) are conjugate in GLn(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.
Al: MonStgElt Default:
Tests whether the finite rational matrix groups G and H are conjugate in GLn(Q). If so, a matrix x such that Gx = 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 GLn(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.