Direct sum decompositions of matroids and exponential structures
type: Item
instance of: scholarly article
title: Direct sum decompositions of matroids and exponential structures
zbMATH Open document ID: 0826.05018
author: Volkmar Welker
published in: Journal of Combinatorial Theory. Series B
publication date: 1995-07-02T00:00:00Z
review text: The author associates to a simple matroid (resp. a geometric lattice) \(M\) and a number \(d\) dividing the rank of \(M\) a partially ordered set \({\mathcal D}_d(M)\) whose upper intervals are (set-) partition lattices. Indeed, for some important cases they are exponential structures in the sense of Stanley. This construction includes the partition lattice, the poset of partitions whose size is divisible by a fixed number \(d\), and the poset of direct sum decompositions of a finite vector space. If \(M\) is a modularly complemented matroid the posets \({\mathcal D}_d(M)\) are CL- shellable. This generalizes results of Sagan and Wachs and settles the open problem of the shellability of the poset of direct sum decompositions. By analyzing the shelling, the author derives some facts about the descending chains and applies these techniques to retrieve the results of Wachs about descending chains in the lattice of \(d\)-divisible partitions. A formula for the Möbius number of the poset of direct sum decompositions of a vector space is also obtained in the paper.
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 767692
zbMATH Keywords: matroid
zbMATH Keywords: geometric lattice
zbMATH Keywords: partially ordered set
zbMATH Keywords: exponential structures
zbMATH Keywords: partition lattice
zbMATH Keywords: poset of partitions
zbMATH Keywords: direct sum decompositions
zbMATH Keywords: finite vector space
zbMATH Keywords: CL-shellable
zbMATH Keywords: shellability
zbMATH Keywords: shelling
zbMATH Keywords: descending chains
zbMATH Keywords: Möbius number
reviewed by: Mirela Ştefănescu
MaRDI profile type: Publication
full work available at URL: jctb.1995.1017
OpenAlex ID: W2035679040
DOI: 10.1006/JCTB.1995.1017
description: scientific article; zbMATH DE number 767692