Coxeter–Petrie Complexes of Regular Maps

Authors: Anderson K.¹; Surowski D.B.²

Source: European Journal of Combinatorics, Volume 23, Number 8, November 2002 , pp. 861-880(20)

Buy & download fulltext article:

Price: $52.63 plus tax (Refund Policy)

Abstract:

Coxeter–Petrie complexes naturally arise as thin diagram geometries whose rank 3 residues contain all of the dual forms of a regular algebraic map M. Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. Jones and Thornton have shown that these involutory duality operations generate the symmetric groupS₃, giving in all six dual forms, and whose source is the outer automorphism group of the infinite triangle group generated by involutions s₁, s₂, s₃, subject to the additional relation s₁s₃ = s₃s₁. In fact, this outer automorphism group is parametrized by the permutations of the three commuting involutions s₁,s₃, s₁s₃. These involutions together with the involutions₂can be taken to define the nodes of a Coxeter diagram of shape D₄(with the involution s₂at the central node), and when the original map M is regular, there is a natural extension from M to a thin Coxeter complex of rank 4 all of whose rank 3 residues are isomorphic to the various dual forms of M. These are fully explicated in case the original algebraic map is a Platonic map. Copyright 2002 Elsevier Science Ltd. All rights reserved.

Language: English

Document Type: Research article

DOI: http://dx.doi.org/10.1006/eujc.2002.0622

Affiliations: 1: Department of Computer Science/Mathematics/Physics, Missouri Western State College, St. Joseph, MO 64507, U.S.A. 2: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, KS 66506-2602, U.S.A.

Publication date: 2002-11-01