Simple Root Systems and Presentations for Certain Complex Reflection Groups

We find all the inequivalent simple root systems for the complex reflection groups G12, G24, G25 and G26. Then we give all the non-congruent essential presentations of these groups by generators and relations. The methed used in the paper is applicable to any finite (complex) reflection groups. Introduction. Shephard and Todd classified all the finite complex reflection groups in paper [5]. Later Cohen gave a more systematic description for these groups in terms of root systems (R, f) and root graphs in [2], in particular, he defined a simple root system (B,w) for the root system of such a group, which is analogous to the corresponding concept for a Coxeter group. In general, for a given finite complex reflection group G, a root system (R, f) is essentially unique but a simple root system (B, w) for (R, f) is not (even up to G-action), e.g., when G = G33, G34 (see [1, Table 4]). One can define an equivalence relation on the set of simple root systems for (R, f) (see 1.7). A natural question is to ask Problem A. How many inequivalent simple root systems are there in total for any irreducible finite complex reflection group G ?