Abstract
The concept of regular incidence complexes generalizes the notion of regular polyhedra in a combinatorial and grouptheoretical sense. A regular (incidence) complex K is a special type of partially ordered structure with regularity defined by the flag-transitivity of its group A(K) of automorphisms.
The structure of a regular complex K can be characterized by certain sets of generators and ‘relations’ of its group. The barycentric subdivision of K leads to a simplicial complex, from which K can be rebuilt by fitting together faces.
Moreover, we characterize the groups that act flag-transitively on regular complexes. Thus we have a correspondence between regular complexes on the one hand and certain groups on the other hand.
Especially, this principle is used to give a geometric representation for an important class of regular complexes, the so-called regular incidence polytopes. There are certain universal incidence polytopes associated to Coxeter groups with linear diagram, from which each regular incidence polytope can be deduced by identifying faces. These incidence polytopes admit a geometric representation in the real space by convex cones.
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Schulte, E. Reguläre Inzidenzkomplexe II. Geom Dedicata 14, 33–56 (1983). https://doi.org/10.1007/BF00182269
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DOI: https://doi.org/10.1007/BF00182269