Diagram rules for the generation of Schubert polynomials
type: Item
instance of: scholarly article
title: Diagram rules for the generation of Schubert polynomials
zbMATH Open document ID: 0917.05082
author: Rudolf Winkel
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-07-18T00:00:00Z
review text: The author considers the Schubert polynomials \(X_{\pi}\in {\mathbb{Z}}[x_1,x_2,\ldots]\) associated with the permutations \(\pi\) contained in the symmetric groups \(S_n\). There are many possible ways to introduce the Schubert polynomials via divided difference operators, recursive generation without divided differences based on the Monk rule and the Bruhat order on permutations, via nil-Coxeter relations, the formula of Billey-Jockusch-Stanley, via sums of mixed shift and multiplication operators, via balanced labeled tableaux, via configurations of labeled pseudo-lines, via flagged Schur modules associated to a Rothe diagram, etc. The theme of the paper is the combinatorial generation of Schubert polynomials via sets of box diagrams. The main reasons to expect a combinatorial rule in terms of box diagrams are: the coefficients in \(X_{\pi}\) are non-negative integers and should count some discrete objects; in the special case of Grassmannian permutations the Schubert polynomial is equal to a Schur function in a finite number of variables and the well-known combinatorial properties of Schur functions should extend to Schubert polynomials. The main result of the paper is the proof of a very elegant and easily applicable combinatorial rule for the generation of Schubert polynomials conjectured in 1990 in the thesis by Kohnert. A similar type of combinatorial rule was given by Bergeron. As an intermediate step in the proof of the Kohnert conjecture the author also obtains a simplified proof of the Bergeron rule. In particular, he shows that the Bergeron rule may be also simplified to a version which is very similar to the recent combinatorial rule of Magyar proved with algebro-geometric methods. The author obtains a direct combinatorial proof of the Magyar rule as well. This shows that there is an algebro-geometric meaning of the Bergeron rule. On the other hand it makes apparent the possibility to give fully combinatorial proofs of other results concerning, e.g., t…
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 13P05
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 1278823
zbMATH Keywords: box diagrams
zbMATH Keywords: diagram rules
zbMATH Keywords: tableaux
zbMATH Keywords: Schubert polynomials
zbMATH Keywords: symmetric functions
zbMATH Keywords: Schur functions
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
full work available at URL: jcta.1998.2931
OpenAlex ID: W1985484328
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DOI: 10.1006/JCTA.1998.2931
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description: scientific article; zbMATH DE number 1278823