A characterization of twin buildings by twin apartments
type: Item
instance of: scholarly article
title: A characterization of twin buildings by twin apartments
zbMATH Open document ID: 0923.51007
DOI: 10.1023/A:1005065400830
author: Peter Abramenko
author: Mark A. Ronan
published in: Geometriae Dedicata
publication date: 1999-10-06T00:00:00Z
review text: Spherical buildings are characterised by their apartments and opposite chambers. In non-spherical buildings, however, one no longer has pairs of opposite chambers. To remedy this situation twin buildings, where chambers in one building can be opposite chambers of another, were introduced. The authors start from the definition of a twin building \((\Delta_+, \Delta_-, \delta^*)\) as given by \textit{J. Tits} [Lond. Math. Soc. Lect. Note Ser. 165, 249-286 (1992; Zbl 0851.22023)] consisting of two buildings \(\Delta_+\) and \(\Delta_-\) of the same type \(M\) and a \(W\)-codistance function \(\delta^*\) defined on \({\mathcal C}_+\times{\mathcal C}_-\cup{\mathcal C}_-\times{\mathcal C}_+\) with values in the Weyl group \(W\). Opposite chambers are chambers of codistance 1. The authors develop a new system of axioms for twin buildings (without the use of a \(W\)-codistance function) more along the lines of Tits' original definition of spherical buildings where the role of spherical apartments is taken by twin apartments. Let \(\Delta=(\Delta_+, \Delta_-,{\mathcal A},\text{op})\) where \(\Delta_+\) and \(\Delta_-\) are two buildings of the same type with respective sets of chambers \({\mathcal C}_+\) and \({\mathcal C}_-\), the set \({\mathcal A}\) of twin apartments is a subset of \(\{(\Sigma_+,\Sigma_-) | \Sigma_\varepsilon\) is an apartment of \(\Delta_\varepsilon, \varepsilon\in\{+,-\}\}\) and \(\text{op}\subseteq {\mathcal C}_+\times{\mathcal C}-\cup{\mathcal C}_-\times{\mathcal C}_+\) is a symmetric opposition relation such that the following four conditions are satisfied: (TA1) For every \(\Sigma=(\Sigma_+,\Sigma_-)\in{\mathcal A}\), the opposition relation induces an isomorphism op\(_\Sigma:\Sigma_+\to\Sigma_- \) between Coxeter complexes. (TA2) For all \(c_+\in{\mathcal C}_+, d_-\in{\mathcal C}_-\), there exists a \(\Sigma\in{\mathcal A}\) such that \((c_+,d_-)\in\Sigma\). (TA3) For all \(\Sigma, \widetilde\Sigma\in{\mathcal A}\) and all \(a=(a_+,a_-)\in\Sigma\c…
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20E42
zbMATH DE Number: 1240983
zbMATH Keywords: twin building
zbMATH Keywords: building
zbMATH Keywords: apartment
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
full work available at URL: a:1005065400830
OpenAlex ID: W3045802
description: scientific article; zbMATH DE number 1240983