Some combinatorial properties of Schubert polynomials
type: Item
instance of: scholarly article
title: Some combinatorial properties of Schubert polynomials
zbMATH Open document ID: 0790.05093
DOI: 10.1023/A:1022419800503
author: Sara Billey
author: William Jockusch
author: Richard P. Stanley
published in: Journal of Algebraic Combinatorics
publication date: 1994-06-22T00:00:00Z
review text: The main result of the Section 1 of the reviewed paper is to give an explicit combinatorial interpretation of the Schubert polynomial \({\mathfrak S}_ w\) in terms of the reduced decompositions of the permutation \(w\). This interpretation is completely different from an earlier conjecture of A. Kohnert and a theorem of N. Bergeron (see \textit{I. G. Macdonald}, Notes on Schubert polynomials, Laboratoire de combinatoire et d'informatique mathematique (LACIM), Univ. du Quebec a Montreal, Montreal, 1991). Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set \(M_ w\) of tableaux with \(w\), each tableau contributing a single term to \({\mathfrak S}_ w\). This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 the authors consider Schubert polynomial \({\mathfrak S}_ w\) when \(w\) has no decreasing subsequence of length three. Such Schubert polynomials have a number of interesting special properties; for instance, they are skew flag Schur (or multi-Schur) functions. In Section 3 they use their results on permutations with no decreasing subsequence of length three to obtain some new combinatorial properties of the rational function \(s_{{\lambda\over\mu}}(1,q,q^ 2,\dots)\), where \(s_{{\lambda\over\mu}}\) denotes a skew Schur function. The authors also formulate several open problems and conjectures.
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 482246
zbMATH Keywords: divided difference operator
zbMATH Keywords: skew flag Schur functions
zbMATH Keywords: Schubert polynomial
zbMATH Keywords: reduced decompositions
zbMATH Keywords: permutation
zbMATH Keywords: Schensted's correspondence
zbMATH Keywords: tableaux
Wikidata QID: Q56625074
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
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full work available at URL: a:1022419800503
OpenAlex ID: W1489169640
description: scientific article; zbMATH DE number 482246