A combinatorial construction of the Schubert polynomials
type: Item
instance of: scholarly article
title: A combinatorial construction of the Schubert polynomials
zbMATH Open document ID: 0771.05097
DOI: 10.1016/0097-3165(92)90002-C
author: Nantel Bergeron
published in: Journal of Combinatorial Theory. Series A
publication date: 1992-09-27T00:00:00Z
review text: Let \(w=(w_ 1,w_ 2,\dots,w_ n)\) be a permutation in the symmetric group \(S_ n\). An explicit combinatorial construction of the Schubert polynomial \({\mathcal S}_ w\) is given as a sum of weights \(x^ D\) of diagrams \(D\) (finite nonempty sets of lattice points \((i,j)\) in the positive quadrant), where the sum is over the set \(\Omega(w)\) of diagrams \(D\) which are obtainable from an initial diagram \(D(w)\) by repeated application of allowable \(B\)-moves. If we view a diagram as a set of checkers (or black stones for those who prefer) distributed over the positive quadrant, then a \(B\)-move is just a move of one checker from a position to an adjacent unoccupied position subject to a set of rules, thereby obtaining a new diagram. Since the classical Schur functions can be viewed as special cases of the Schubert polynomials for appropriate choices of the permutation \(w\), it has been hoped that there was a combinatorial algorithm for constructing Schubert polynomials which extends (or is at least in analogy to) the construction of the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\) as a sum of weights of column strict tableaux of shape \(\lambda\), where \(\lambda\) is a partition. The algorithm given here is not quite an extension of the usual construction for Schur functions, however a simple bijection between \(\Omega(w)\) and the set of column strict tableaux of shape \(\lambda\) is given in the particular case where the Schubert polynomial \({\mathcal S}_ w\) is the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\). A different algorithm for constructing the general Schubert polynomial \({\mathcal S}_ w\) was conjectured (and proved when \(w\) is a vexillary permutation) by \textit{A. Kohnert} [Weintrauben, Polynome, Tableaux, Bayreuther Math. Schr. 38, 1-97 (1991; Zbl 0755.05095)]. A sketch is given at the end of the paper here of how one would show the equivalence of the rule given here and that of Kohnert.
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 64857
zbMATH Keywords: symmetric group
zbMATH Keywords: Schubert polynomial
zbMATH Keywords: Schur functions
zbMATH Keywords: tableaux
zbMATH Keywords: algorithm
reviewed by: Robert A. Gustafson
MaRDI profile type: Publication
cites work: Sergeev's formula and the littlewood—richardson rule
cites work: SCHUBERT CELLS AND COHOMOLOGY OF THE SPACESG/P
cites work: Désingularisation des variétés de Schubert généralisées
cites work: Q3978180
cites work: Puissances extérieures, déterminants et cycles de Schubert
cites work: Q3958581
cites work: Q3958580
cites work: Q3329580
cites work: Q3329581
cites work: Schubert polynomials and the Littlewood-Richardson rule
cites work: Q3709222
cites work: Symmetrization operators on polynomial rings
cites work: Q3947818
cites work: Q3975127
cites work: Flagged Schur functions, Schubert polynomials, and symmetrizing operators
description: scientific article; zbMATH DE number 64857