Whitney homology of semipure shellable posets
type: Item
instance of: scholarly article
title: Whitney homology of semipure shellable posets
zbMATH Open document ID: 0922.06005
DOI: 10.1023/A:1018694401498
author: Michelle L. Wachs
published in: Journal of Algebraic Combinatorics
publication date: 1999-10-05T00:00:00Z
review text: Group representations on homologies of posets have been the object of intensive research in the past few decades. Usually posets are expected to be Cohen-Macaulay or shellable, since then only the top homology is nontrivial so that the result follows from the routine application of the Hopf trace formula. The situation becomes much more complicated if a poset \(P\) has nonvanishing homology in various dimensions, as it could occur, for example, for nonpure \(P\). In the paper under review the author applies the theory of nonpure shellability (developed by \textit{A. Björner} and \textit{M. L. Wachs} [Trans. Am. Math. Soc. 348, 1299-1327 (1996; Zbl 0857.05102) and ibid. 349, 3945-3975 (1997; Zbl 0886.05126)]) to computing representations of the symmetric group on each homology for a poset of partitions whose block sizes are congruent to \(k \text{mod} d\) for any \(k\) and \(d\). The cases \(k=0\) and \(k=1\) (when the posets are pure shellable) were considered by \textit{A. R. Calderbank, P. Hanlon} and \textit{R. W. Robinson} [Proc. Lond. Math. Soc., III. Ser. 53, 288-320 (1986; Zbl 0602.20017)]. However, for general \(k\) the \(k \text{mod} d\) partition poset is not pure! An application to subspace arrangements is also discussed.
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 1275699
zbMATH Keywords: poset homology
zbMATH Keywords: shellability
zbMATH Keywords: plethysm
zbMATH Keywords: symmetric group
zbMATH Keywords: partition lattice
zbMATH Keywords: partition poset
zbMATH Keywords: subspace arrangements
reviewed by: Valeriy B. Mnukhin
MaRDI profile type: Publication
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full work available at URL: a:1018694401498
OpenAlex ID: W1584726502
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description: scientific article; zbMATH DE number 1275699