Posets, regular CW complexes and Bruhat order
type: Item
instance of: scholarly article
title: Posets, regular CW complexes and Bruhat order
zbMATH Open document ID: 0538.06001
DOI: 10.1016/S0195-6698(84)80012-8
published in: European Journal of Combinatorics
publication date: 1984-01-01T00:00:00Z
review text: A poset P is thin if all its intervals of length two have cardinality four. For other terminology see e.g., the author and \textit{M. Wachs} [Trans. Am. Math. Soc. 277, 323-341 (1983; Zbl 0514.05009)]. In this very interesting paper the author continues his efforts at the classification of those posets which are somehow (combinatorially) topological in nature. In this case the posets are the ones obtained from CW-complexes K. Indeed, if K is such a complex then F(K) is the set of closed cells ordered by containment and augmented by a bottom element \(\hat0\). A d-CW- complex is a finite regular CW-complex such that every closed cell is a face of a d-dimensional cell. Shellability is defined for d-CW-complexes. The author demonstrates that if P is a thin, nontrivial, finite, pure poset of length \(d+1\) with least element \(\hat0\), then P shellable \(\Rightarrow\) \(P\cong F(K)\), K a d-CW-complex; \^P\(=P\cup \{\hat 0\}\) dual CL-shellable \(\Leftrightarrow\) K can be a shellable d-CW-complex; \^P shellable and thin \(\Rightarrow\) K can be a d-CW-complex homeomorphic to the d-sphere; \^P dual shellable and thin \(\Leftrightarrow\) K can be a shellable d-CW- complex homeomorphic to the d-sphere. Using these results and observations applied to Coxeter groups and Bruhat orderings, the author sharpens some of his own results. He demonstrates that every Coxeter group (W,S) determines a topological space with a regular cell decomposition such that the Bruhat ordering of W is isomorphic to the incidence ordering of the cells. The associated space is a sphere when W is finite and contractible when W is infinite.
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 57Q05
Mathematics Subject Classification ID: 57N99
Mathematics Subject Classification ID: 52Bxx
zbMATH DE Number: 3855170
zbMATH Keywords: thin posets
zbMATH Keywords: face posets
zbMATH Keywords: closed cells
zbMATH Keywords: finite regular CW-complex
zbMATH Keywords: pure poset
zbMATH Keywords: shellable d-CW-complex
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Bruhat orderings
author: Anders Bjoerner
reviewed by: J. Neggers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3855170