Alexis Suki Dasher: Solvable lattice models for Kirillov’s universal divided difference operators

Speaker
Alexis Suki Dasher,

Abstract
Divided difference operators are well-known for their role in computing Schubert classes in the cohomology of flag varieties, which are represented by the Schubert polynomials. Minor modifications to these operators compute various other geometric data and functions, such as Demazure characters and \(\beta\)-Grothendieck polynomials. To unify these modifications, Anatol Kirillov introduced a \(5\)-parameter family of divided difference operators which are essentially universal among generalizations of type-\(A\) divided difference operators satisfying the braid relations.

We present a family of solvable lattice models whose partition functions form a vast collection of polynomials computed by Kirillov’s generalized divided difference operators. As a consequence, we provide a uniform framework for computing numerous families of familiar functions, recover existing lattice models as specializations of our construction, and as an application, we resolve Kirillov’s conjecture that the coefficients of the Hecke–Grothendieck polynomials are non-negative.