Classification of crystallographic groups associated with Coxeter groups

Abstract

This paper is concerned with the structure of those space groups (or crystallographic groups) in n-dimensional Euclidean space for which the point groups are essential and generated by reflections. Recall [ 1 ] that a subgroup G in GL( V) is said to be essential if VG = (0). Those point groups are Weyl groups of root systems in the vector space [l]. The fundamental results are presented in Section 2. In a first step, we restrict our study to irreducible root systems. The principal result is stated in Theorem 2.6: in most cases the group H’(G, V/P(R)) is zero if the root system R is an irreducible one and the point group G is the Weyl group of R. Then, an easy consequence is given in Theorem 2.11: the same result holds even if the root system is a reducible one: we just have to forbid some types for its components. If /i is an invariant lattice such that Q(R) c /1 c P(R) the short exact sequence