Some combinatorial properties of Schubert polynomials. (English) Zbl 0790.05093

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 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.


05E05 Symmetric functions and generalizations
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