Reflection Groups and Polytopes over Finite Fields, III

2007/07/26 by Barry Monson, Monson, Barry, Egon Schulte +1
Computer Science · Mathematics · #20F55 (Secondary) #51M20 (Primary) #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Metric Geometry (math.MG) #math.CO #math.MG #msc:20F55 #msc:51M20 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0707.4007

Advances in Applied Mathematics (to appear); 19 pages

arxiv created 2007/07/26 · openalex publication_date 2007/07/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

When the standard representation of a crystallographic Coxeter group is reduced modulo an odd prime p, one obtains a finite group G^p acting on some orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to groups of rank at most 4, as well as all groups of spherical or Euclidean type. Here we extend the range of our earlier criteria for the polytopality of G^p . Building on this we investigate the class of 3-infinity groups of general rank, and then complete a survey of those locally toroidal polytopes which can be described by our construction.

Related

• Reflection groups and polytopes over finite fields, II
• Locally Toroidal Polytopes and Modular Linear Groups
• A family of regular polytopes of order 4p^m with type \p, 2p\
• Polytopes with groups of type PGL₂(q)
• Groups of type L₂(q) acting on polytopes