Congruence classes of presentations for the complex reflection groups G( m, 1, n) and G( m, m, n)

Abstract

In the present paper, we give a graph-theoretic description for representatives of all the congruence classes of presentations (or r.c.p. for brevity) for the imprimitive complex reflection groups G( m, 1, n) and G( m, m, n). We have three main results. The first main result is to establish a bijection between the set of all the congruence classes of presentations for the group G( m, 1, n) and the set of isomorphism classes of all the rooted trees of n nodes. The next main result is to establish a bijection between the set of all the congruence classes of presentations for the group G( m, m, n) and the set of isomorphism classes of all the connected graphs with n nodes and n edges. Then the last main result is to show that any generator set S of G = G( m, 1, n) or G( m, m, n) of n reflections, together with the respective basic relations on S, form a presentation of G.