Reflection groups and polytopes over finite fields, II

Details Reflection groups and polytopes over finite fields, II Reflection groups and polytopes over finite fields, II

2006-01-20

arxiv.org/abs/math/0601502v1

Mathematics

Metric Geometry

30 pages (Advances in Applied Mathematics, to appear)

Scientific paper

When the standard representation of a crystallographic Coxeter group $\Gamma$ is reduced modulo an odd prime $p$, a finite representation in some orthogonal space over $\mathbb{Z}_p$ is obtained. If $\Gamma$ has a string diagram, the latter group will often be the automorphism group of a finite regular polytope. In Part I we described the basics of this construction and enumerated the polytopes associated with the groups of rank 3 and the groups of spherical or Euclidean type. In this paper, we investigate such families of polytopes for more general choices of $\Gamma$, including all groups of rank 4. In particular, we study in depth the interplay between their geometric properties and the algebraic structure of the corresponding finite orthogonal group.

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