Workshop on Combinatorics of Enumerative Geometry

Abstract: In his 2018 paper "Some Schubert shenanigans" Richard Stanley asked for the asymptotic behavior of the maximal principle specialization of a Schubert polynomial. Motivated by this, still open, question we explore the generalization to Grothendieck polynomials and uncover a rich statistical mechanical structure giving new permutons. The models via pipe dreams are tiling models, which can also be viewed as ensembles of lattice walks with various interaction constraints. We understand the typical Grothendieck permutations via the theory of the Totally Asymmetric Simple Exclusion Process (TASEP), which itself is analyzed by Schur functions. Deeper connections with free fermion 6 vertex models and domino tilings of the Aztec diamond allow us to describe the extreme cases of the original algebraic problem and to answer the question which permutations maximize the principle specialization of the Grothendieck polynomials at $\beta=1$. Based on joint work with A. H. Morales, L. Petrov, D. Yeliussizov.