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We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family $G(m, p, n)$ of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive and give a simple criterion to tell when two shortest reflection factorizations belong to the same Hurwitz orbit. We also characterize the quasi-Coxeter elements (those with a shortest reflection factorization that generates the whole group) in $G(m, p, n)$.Mathematics Subject Classifications: 05A05, 05A15, 05E18, 20F55

We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family \(G(m, p, n)\) of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive and give a simple criterion to tell when two shortest reflection factorizations belong to the same Hurwitz orbit. We also characterize the quasi-Coxeter elements (those with a shortest reflection factorization that generates the whole group) in \(G(m, p, n)\).Mathematics Subject Classifications: 05A05, 05A15, 05E18, 20F55

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either G(r,p,n) with gcd(p,n)=1; G(r,p,2) with r/p odd; or G23, the Coxeter group of type H3. We additionally provide explicit formulas for all automorphisms of G(r,p,n), and construct new Gelfand models for the groups G(r,p,n) with gcd(p,n)=1. © 2011 Elsevier Inc.

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either G(r,p,n) with gcd(p,n)=1; G(r,p,2) with r/p odd; or G23, the Coxeter group of type H3. We additionally provide explicit formulas for all automorphisms of G(r,p,n), and construct new Gelfand models for the groups G(r,p,n) with gcd(p,n)=1. © 2011 Elsevier Inc.

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either G(r,p,n) with gcd(p,n)=1; G(r,p,2) with r/p odd; or G23, the Coxeter group of type H3. We additionally provide explicit formulas for all automorphisms of G(r,p,n), and construct new Gelfand models for the groups G(r,p,n) with gcd(p,n)=1. © 2011 Elsevier Inc.

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either G(r,p,n) with gcd(p,n)=1; G(r,p,2) with r/p odd; or G23, the Coxeter group of type H3. We additionally provide explicit formulas for all automorphisms of G(r,p,n), and construct new Gelfand models for the groups G(r,p,n) with gcd(p,n)=1. © 2011 Elsevier Inc.

We give a model of the coinvariant algebra of the complex reflection groups as a subalgebra of a braided Hopf algebra called Nichols–Woronowicz algebra.