Regular polytopes from twisted Coxeter groups and unitary reflexion groups
type: Item
instance of: scholarly article
title: Regular polytopes from twisted Coxeter groups and unitary reflexion groups
zbMATH Open document ID: 0711.51007
DOI: 10.1016/0001-8708(90)90083-Y
author: Peter McMullen
author: Egon Schulte
published in: Advances in Mathematics
publication date: 1990-01-01T00:00:00Z
review text: In several papers the authors studied the following ``amalgamation problem''. Given two abstract regular d-polytopes \(P_ 1\) and \(P_ 2\) such that the vertex figures of \(P_ 1\) are isomorphic to the facets of \(P_ 2\), does there exist a regular \((d+1)\)-polytope P with facets isomorphic to \(P_ 1\) and vertex-figures isomorphic to \(P_ 2?\) The universal member among all solutions (if it exists) is denoted by \(\{P_ 1,P_ 2\}\). Several techniques have been developed to decide for some interesting cases the existence and finiteness or non-finiteness of \(\{P_ 1,P_ 2\}\). Twisting operations on Coxeter groups have been introduced by the authors in Math. Z. 201, 209-226 (1989; Zbl 0646.51023), as a method to solve the amalgamation problem. In the present paper the particular case is studied where the polytopes are obtained from unitary reflexion groups generated by reflexions of period 2.
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 4169552
zbMATH Keywords: symmetry group
zbMATH Keywords: realization of an abstract polytope
zbMATH Keywords: semi-direct product
zbMATH Keywords: twisted Coxeter groups
zbMATH Keywords: regular d-polytopes
zbMATH Keywords: amalgamation problem
zbMATH Keywords: unitary reflexion groups
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
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full work available at URL: 0001-8708(90)90083-y
OpenAlex ID: W2002387862
description: scientific article; zbMATH DE number 4169552