Skew Divided Difference Operators and Schubert Polynomials

TL;DR

This paper explores the action of skew divided difference operators on Schubert polynomials, providing explicit formulas for their structural constants and conditions for positivity of coefficients.

Contribution

It introduces an explicit formula for structural constants of Schubert polynomials using weighted paths in Bruhat order and establishes positivity conditions for transformed polynomials.

Findings

Explicit formula for structural constants in Schubert polynomials

Conditions under which skew divided difference operators produce positive coefficients

Connection between operators and weighted paths in Bruhat order

Abstract

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.