Some combinatorial properties of Schubert polynomials

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

Journal of Algebraic Combinatorics, Tome 2 (1993) no. 4, pp. 345-374.

Résumé

Summary: 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 $\mathfrak S _{ w}$ mathfrakS_$\omega $ 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 $M _{ w}$ mathcalM_$\omega $ of tableaux with $w$, each tableau contributing a single term to $\mathfrak S _{ w}$ mathfrakS_$\omega $ . 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 $\mathfrak S _{ w}$ mathfrakS_$\omega $ is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function $8 _{ l/ m} (1, q, q ^{2}$ , frac14 ) 8_$\lambda /\mu $ (1,q,q^2 , $\ldots $) , where $8 _{ l/ m}$ 8_$\lambda /\mu $ denotes a skew Schur function.

Détail

Keywords: divided difference operator, Schubert polynomial, reduced decomposition, edelman-Greene correspondence, 321-avoiding permutation, flag skew Schur function, principal specialization