Imprimitive Unitary Reflection Groups

Let B be the direct product of n copies of the cyclic group C_m of order m and represent the elements of B by diagonal matrices diag(theta_1, theta_2, ..., theta_n). The elements of the symmetric group Sym(n) can be represented by n x n permutation matrices and in this guise it acts on the group B; the resulting semidirect product is also known as the wreath product C_m wreath Sym(n).

For each divisor p of m define

A(m, p, n) := { (theta_1, theta_2, ..., theta_n) in B | (theta_1theta_2.stheta_n)^(m/p) = 1 }.

It is immediately clear that A(m, p, n) is a subgroup of index p in B that is invariant under the action of Sym(n). The semidirect product of A(m, p, n) by the symmetric group Sym(n) is the group G(m, p, n). Shephard and Todd proved that every irreducible imprimitive unitary reflection subgroup of GL(n, C) is conjugate to G(m, p, n) for some m and p.

ImprimitiveReflectionGroup(m, p, n) : RngIntElt, RngIntElt, RngIntElt -> GrpMat, Fld

This function returns the Shephard-Todd group G(m, p, n), where p divides m. The field of definition is returned as a second value. In general, G(m, p, n) is irreducible but if m = p = 1, the function returns Sym(n) in its natural permutation representation, which is not irreducible.

ImprimitiveReflectionGroup(GrpFP,{ m, p, n}) : Cat, RngIntElt, RngIntElt, RngIntElt -> GrpFP

Construct the imprimitive complex reflection group G(m, p, n) as finitely presented group, given by its standard presentation[Rou98].