Quantum K-structure constants and cylindric Grothendieck polynomials for Grassmannians

Abstract

We use the partition functions of a five-vertex exactly solvable lattice model from statistical mechanics to give new combinatorial formulae for the Schubert structure constants of torus-equivariant quantum K-theory rings of Grassmannians. The proofs build on our earlier results where we identified the transfer matrices of these lattice models geometrically as multiplications by the Hirzebruch $λ_y$ classes of the tautological subbundle and the (dual) quotient bundle. The structure constants are solutions of a system of linear equations with coefficients given by the partition function. This system may be solved explicitly by specializing the Hirzebruch parameter $y$ to an equivariant parameter, leading to explicit formulae using Cramer's rule. Furthermore, we give an alternative combinatorial construction of the partition functions in terms of sums over cylindric set-valued tableaux, which we call cylindric Grothendieck polynomials. The latter generalise McNamara's factorial skew Grothendieck and Yeliussizov's modified Grothendieck polynomials and can be thought of as an equivariant K-theoretic version of Postnikov's toric Schur function. We also provide new polynomial representatives for equivariant K-theory classes of Richardson varieties and their variants.