Isomorphisms, automorphisms, and generalized involution models of projective reflection groups

We investigate the generalized involution models of the projective reflec- tion groups G(r, p, q, n). This family of groups parametrizes all quotients of the complex reflection groups G(r, p, n) by scalar subgroups. Our clas- sification is ultimately incomplete, but we provide several necessary and sufficient conditions for generalized involution models to exist in various cases. In the process we solve several intermediate problems concerning the structure of projective reflection groups. We derive a simple criterion for determining whether two groups G(r, p, q, n) and G(r, p , q , n) are iso- morphic. We also describe explicitly the form of all automorphisms of G(r, p, q, n), outside a finite list of exceptional cases. Building on prior work, this allows us to prove that G(r, p, 1, n) has a generalized involution model if and only if G(r, p, 1, n) and G(r, 1, p, n) are isomorphic. We also classify which groups G(r, p, q, n) have generalized involution models when n = 2, or q is odd, or n is odd.