Dominic Searles: Schubert polynomials and slide polynomials

Abstract

We introduce a new basis for the polynomial ring called the slide polynomials, which contains Gessel's fundamental basis of quasisymmetric polynomials. One aim is to better understand the geometrically-important basis of Schubert polynomials, whose structure constants count intersection points of triples of Schubert subvarieties of the complete flag variety. Schubert polynomials expand positively in the slide basis, generalizing Gessel's expansion of Schur polynomials in the fundamental basis of quasisymmetric polynomials. The slide basis moreover has positive structure constants; we give a combinatorial Littlewood-Richardson rule for these numbers in terms of shuffles. We also introduce another basis for polynomials which contains the quasi-Schur basis of quasisymmetric polynomials, and explore its relationships with other bases. (Joint work with S. Assaf.)