Finding combinatorial interpretations for the structure constants of Schubert polynomials is a long-standing open problem in algebraic combinatorics. In the case where one of the Schubert polynomials is a Schur polynomial, the structure constants are encoded in a poset called the “k-Bruhat order”. In studying the k-Bruhat order, Bergeron and Sottile were led to introduce a monoid which encodes the chain structure of the k-Bruhat order. Using the monoid structure, the authors were able to establish properties and descriptions of certain structure constants. In this talk, after outlining the developments discussed above, we discuss ongoing work concerning an analogous story for quantum Schubert polynomials and an associated quantum k-Bruhat order. This is joint work with Laura Colmenarejo.

Speaker’s Website: https://sites.google.com/view/nmayers