Some Combinatorial Properties of Schubert Polynomials

  • Authors:

  • Sara C. Billey;William Jockusch;Richard P. Stanley

  • Affiliations:

  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139. Supported by the National Physical Science Consortium. Current address: Department of Mathematics, UCS ...;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139. Supported by an NSF Graduate Fellowship. Current address: Department of Mathematics, University of Mic ...;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139. Partially supported by NSF grants DMS-8901834 and DMS-9206374

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 1993

Quantified Score

Hi-index 0.01

Visualization

Abstract

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial {\mathfrak S}_w in terms of the reduced decompositions of the permutation w. 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 {\cal 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 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that {\mathfrak S}_w is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function s_{\lambda/\mu}(1,q,q^2,\cdots), where s_{\lambda/\mu} denotes a skew Schur function.