The crystallography of Coxeter groups

Abstract

This paper is primarily concerned with the structure of those space groups in n-dimensional Euclidean space for which the point groups are generated by reflections, i.e., are crystallographic Coxeter groups. The determination of all possible space groups dates back to the last century for n < 3 and has recently been extended to n = 4 by Brown, Neubiiser and Zassenhaus [3-51 with the aid of a computer and some previous work by Dade [7]. However, since eventually every finite group will occur as the point group of some space group, one cannot expect a reasonable solution to the problem for a general n. On the other hand, by restricting oneself to an accessible class of point groups, such as we have chosen, one can hope to obtain a satisfactory answer. The interest in Coxeter point groups is also heightened by the fact that, together with their subgroups, such groups are sufficient for n sl 3. Our discussion can thus form the basis of a less ad hoc exposition than the one usually employed in that case [6]. A sequel [ 1 I] dealing with the systematic treatment of the case when the point group is the rotation subgroup of a Coxeter group will appear soon. In Section 1, we present a general discussion of the problems of mathematical crystallography in affine space, including in particular a generalization of some results of Zassenhaus [lo]. This is extended in Section 2 in the special case of Euclidean space. The principal results are presented in Section 3 and then applied in Section 4 to the classical situation. We also include an interpretation in our terms of the maximal groups found by Dade [7] in the case n = 4 and Rygkov [9] in the case n = 5. After this was written, a paper of Schwarzenberger [12] appeared containing, among other things, some further results in the special case c, x .‘. x c,.