The center conjecture for non-exceptional buildings
type: Item
instance of: scholarly article
title: The center conjecture for non-exceptional buildings
zbMATH Open document ID: 1101.51004
author: Bernhard Mühlherr
author: Jacques Tits
published in: Journal of Algebra
publication date: 2006-08-16T00:00:00Z
review text: The centre conjecture for buildings states that a convex subcomplex \(\tilde\Delta\subset\Delta\) of a spherical building \({\pmb\Delta}=(\Delta,\subset)\) is completely reducible, that is, for each simplex \(A\in \tilde\Delta\) there is a simplex in \(\tilde\Delta\) which is opposite to \(A\) in \({\pmb\Delta}\), or is in the centre, that is, the stabilizer of \(\tilde\Delta\) in the automorphism group of \({\pmb\Delta}\) fixes a non-trivial simplex of \(\tilde\Delta\). The main result of the paper under review is that the centre conjecture holds for irreducible spherical buildings which are not of type \(E_6\), \(E_7\), \(E_8\) or \(F_4\). This then carries over to spherical buildings whose types have no direct factors \(E_6\), \(E_7\), \(E_8\) or \(F_4\). The proof of the main result follows ideas developed in [\textit{J.-P. Serre}, Astérisque 299, 195--217 (2005; Zbl 1156.20313)] and \textit{J. Tits} [Résumé de cours, in: Annuaire du Collège de France, 97e année, 89--102 (1996-97)]. It relies on the fact that buildings of finite rank are flag complexes of certain classes of incidence geometries, that is, generalized polygons, projective spaces and polar spaces, for which the authors derive combinatorial properties and give characterizations of opposite vertices in these buildings and of convex subcomplexes by closedness. Using an associated building of type \(F_4\) a proof of the centre conjecture for buildings of type \(D_4\) is outlined since this special case requires additional arguments as trialities may exist so that the automorphism group of such a building may be bigger.
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 5047184
zbMATH Keywords: spherical building
zbMATH Keywords: convex subset
zbMATH Keywords: generalized polygon
zbMATH Keywords: projective space
zbMATH Keywords: polar space
MaRDI profile type: Publication
Wikidata QID: Q122916931
full work available at URL: j.jalgebra.2006.01.011
OpenAlex ID: W2092450758
cites work: On the foundations of polar geometry. II
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cites work: Buildings of spherical type and finite BN-pairs
cites work: Q3961400
cites work: Q4779974
DOI: 10.1016/J.JALGEBRA.2006.01.011
description: scientific article