Kirillov's conjecture on Hecke--Grothendieck polynomials, A. Suki Dasher (University of Minnesota) -- December 12th, 2024

A. N. Kirillov introduced a family of multi-parameter polynomials which are vast generalizations of the Schubert polynomials. They are recursively defined by the largest class of divided difference operators satisfying the braid relations, and their specializations include the Schubert and Grothendieck polynomials, among others. We construct a new family of solvable lattice models whose partition functions are defined by the "generic" subfamily of Kirillov's operators and which specialize to Kirillov's polynomials. As a consequence, we prove Kirillov's conjecture that the subfamily of so-called Hecke–Grothendieck polynomials have non-negative coefficients, and we demonstrate that the larger family of polynomials exhibits rare instances of negative coefficients. This is joint work with Ben Brubaker, Michael Hu, Nupur Jain, Yifan Li, Yi Lin, Maria Mihaila, Van Tran, and I. Deniz Ünel.

The seminar co-organizers are Ian Cavey, Andrew Hardt, David Keating, and Alexander Yong.