An insertion algorithm for multiplying Demazure characters by Schur polynomials

Sami H. Assaf

Published 2021-09-13Version 1

Certain polynomial generalizations of Schur polynomials, such as Demazure characters and Schubert polynomials, have structure constants with geometric significance, thus motivating the search for combinatorial formulas for these numbers. In this paper, we introduce an insertion algorithm on Kohnert's combinatorial model for these polynomials, generalizing Robinson--Schensted--Knuth insertion on tableaux. This new insertion algorithm yields an explicit, nonnegative formula expressing the product of a Demazure character and a Schur polynomial as a sum of Schubert characters, partially resolving a conjecture of Polo. Moreover, we lift this expression to a nonnegative, combinatorial formula for the Demazure character expansion of the product of a Schubert polynomial and a Schur polynomial providing new progress toward a combinatorial formula for Schubert structure constants.