A simple definition for the universal Grassmannian order
type: Item
instance of: scholarly article
title: A simple definition for the universal Grassmannian order
zbMATH Open document ID: 1018.05107
DOI: 10.1016/S0097-3165(03)00047-5
author: Curtis D. Bennett
author: Lakshmi Evani
author: David Grabiner
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-07-01T00:00:00Z
review text: This paper is devoted to a simple combinatorial description of a partial order, the Grassmannian-Bruhat order (\(<\)) on the symmetric group \(S_n\) introduced by \textit{N. Bergeron} and the reviewer to study the problem of multiplying Schubert polynomials [Duke Math. J. 95, 373-423 (1998; Zbl 0939.05084)]. Its main results include a new and simpler definition for this order which characterizes it as as the meet in the poset of partial orders on \(S_n\) of the usual weak order and one given by a certain permutation statistic, which turns out to be the rank function for \(<\). Most properties of \(<\) (including those important for loc. cit.) follow quite directly from this characterization, simplifying the original arguments. The final section studies the rank-generating function for \(<\), showing that it is divisible by \((1+t)^{\lfloor\frac{n}{2}\rfloor}\). Two early typos (both omitting `\(>\)') may impede the understanding of the reader: The end of line 8 of Section 1 (p.~348) should be `and \(\sigma[i]>\sigma[j]\), and'. Also, the right hand side of first displayed formula on page 349 should be `\(\{i\mid \sigma[i]>i\}\)'.
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1945002
zbMATH Keywords: Grassmannian-Bruhat order
zbMATH Keywords: Schubert polynomials
zbMATH Keywords: poset of partial orders
zbMATH Keywords: permutation statistic
reviewed by: Frank Sottile
MaRDI profile type: Publication
cites work: Schubert polynomials, the Bruhat order, and the geometry of flag manifolds
cites work: A Pieri-type formula for isotropic flag manifolds
cites work: A monoid for the Grassmannian Bruhat order
cites work: Q3998721
cites work: Q3748279
full work available at URL: s0097-3165(03)00047-5
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description: scientific article; zbMATH DE number 1945002