CONGRUENCE CLASSES OF PRESENTATIONS FOR THE COMPLEX REFLECTION GROUPS G(m, 1, n) AND G(m,m, n)

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. In the paper [5], I introduced two concepts for any complex reflection group G generated by more than two reflections: one is the equivalence of simple root systems, and the other is the congruence of presentations. According to the definition, the equivalent simple root systems of G determine the congruent presentations of G. Then I, together with my students, Wang Li and Zeng Peng, found all the inequivalent simple root systems for all the primitive complex reflection groups except the group G34 (see [5][8][9]). We also described explicitly r.c.p. for the groups G12, G24, G25, G26, G7, G15, G27. Then 1991 Mathematics Subject Classification. 20F55.