On the expansion of Schur and Schubert polynomials into standard elementary monomials
type: Item
instance of: scholarly article
title: On the expansion of Schur and Schubert polynomials into standard elementary monomials
zbMATH Open document ID: 0920.05069
author: Rudolf Winkel
published in: Advances in Mathematics
publication date: 1999-09-15T00:00:00Z
review text: Any polynomial in \(\mathbb{Z}[x]\) can be expressed in terms of elementary symmetric polynomials which can in turn be expressed as linear combinations of standard elementary monomials or SEM. The paper under review examines the SEM expansions of Schur and Schubert polynomials. The SEM expansion for Schubert polynomials is of particular interest because quantum Schubert polynomials can be computed by quantizing the SEM expansion of ordinary Schubert polynomials [\textit{S. Fomin}, \textit{S. Gelfand} and \textit{A. Postnikov}, Quantum Schubert polynomials, J. Am. Math. Soc. 10, No. 3, 565-596 (1997; Zbl 0912.14018)]. In the case of Schur functions, the SEM expansion can be obtained via a variant of the Jacobi-Trudi identity, and the author demonstrates how a combinatorial rule based on posets of staircase box diagrams can be used to calculate SEM expansion coefficients. In the case of Schubert polynomials, the SEM expansion can be obtained via the determinantal expression of \textit{A. N. Kirillov} and \textit{T. Maeno} [Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, q-alg/9610022, 1996, preprint, 52 pp.] and the author conjectures a combinatorial rule similar to that proved for Schur functions.
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 1215516
zbMATH Keywords: elementary symmetric polynomials
zbMATH Keywords: standard elementary monomials
zbMATH Keywords: Schubert polynomials
zbMATH Keywords: SEM expansion
zbMATH Keywords: Schur functions
reviewed by: Angèle. M. Hamel
MaRDI profile type: Publication
full work available at URL: aima.1998.1730
OpenAlex ID: W1970322938
cites work: SCHUBERT CELLS AND COHOMOLOGY OF THE SPACESG/P
cites work: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts
cites work: Invariants symétriques entiers des groupes de Weyl et torsion
cites work: Désingularisation des variétés de Schubert généralisées
cites work: Quantum Schubert polynomials
cites work: Q3998721
cites work: Q3942064
cites work: QUANTUM SCHUBERT POLYNOMIALS AND QUANTUM SCHUR FUNCTIONS
cites work: Q4505506
cites work: Décompositions dans l'algèbre des différences divisées. (Decompositions in the algebra of divided differences)
cites work: Q3473264
cites work: Q3975127
cites work: Q3947818
cites work: Q4001520
cites work: Recursive and combinatorial properties of Schubert polynomials
cites work: Diagram rules for the generation of Schubert polynomials
DOI: 10.1006/AIMA.1998.1730
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description: scientific article; zbMATH DE number 1215516