Deformation of chains via a local symmetric group action
type: Item
instance of: scholarly article
title: Deformation of chains via a local symmetric group action
zbMATH Open document ID: 0921.05062
author: Patricia Hersh
published in: The Electronic Journal of Combinatorics
publication date: 1999-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/119919
full work available at URL: EJC
review text: A symmetric group action on the maximal chains in a finite, ranked poset \(P\) was defined by \textit{R. P. Stanley} [Electron. J. Comb. 3, No. 2, Research paper R6 (1996; Zbl 0857.05091); printed version J. Comb. 3, No. 2, 161-182 (1996)] to be local if for each \(i\) the transposition \((i,i+1)\) sends each maximal chain either to itself or to one differing only at rank \(i\). A prototypical example is the natural action of \(S_n\) on the maximal chains of the Boolean algebra \({\mathcal B}_n\). Here the only orbit is \({\mathcal B}_n\) itself, which is a product of 2-chains. The main result of the paper under review says that this is related only with the fact that \({\mathcal B}_n\) is a lattice: each orbit under a local action of \(S_n\) on the maximal chains of a lattice is a product of chains. For a general poset \(P\) it is shown that the Frobenius characteristic of a local action of \(S_n\) on the maximal chains of \(P\) is an \(h\)-positive symmetric function. Some applications to symmetric chain decompositions of posets and to noncrossing partition lattices are also discussed.
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1309970
zbMATH Keywords: local action
zbMATH Keywords: orbit
zbMATH Keywords: maximal chain
zbMATH Keywords: symmetric group
zbMATH Keywords: partition lattice
reviewed by: Valeriy B. Mnukhin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1309970