On Coxeter diagrams of complex reflection groups. (Q2841375)

The finite complex reflection groups have been classified by \textit{G. C. Shephard} and \textit{J. A. Todd} [Can. J. Math. 6, 274-304 (1954; Zbl 0055.14305)]; see also \textit{A. M. Cohen} [Ann. Sci. Éc. Norm. Supér. (4) 9, 379-436 (1976; Zbl 0359.20029)], \textit{G. I. Lehrer} and \textit{D. E. Taylor} [Unitary reflection groups. Aust. Math. Soc. Lect. Ser. 20. Cambridge: Cambridge University Press (2009; Zbl 1189.20001)]. The author gives a new proof for the classification of all root lattices over the Eisenstein integers \(\mathbb Z[e^{2\pi i/3}]\). Moreover, he considers the problem to associate diagrams with complex reflection groups (resembling Coxeter diagrams for real reflection groups); he proposes an algorithm and reports on some computer experiments. For some groups, new diagrams are obtained. This leads to presentations for the corresponding braid groups.