Speaker
Greta Panova, University of Southern California
Abstract
Full title: “Asymptotics and maxima for Schubert and Grothendieck polynomials: from algebra to probability via vertex models”
In his 2018 paper “”Some Schubert shenanigans,”” Richard Stanley asked for the asymptotic behavior of the maximal principal specialization of a Schubert polynomial. Motivated by this (still open) question, we explore the generalization to Grothendieck polynomials and uncover a rich statistical mechanics 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 principal specialization of the Grothendieck polynomials at beta=1.
Back to the original question on Schubert polynomials, we explore the realm computationally, revealing similar behavior to the Grothendieck case. We also disprove the Merzon-Smirnov conjecture that the maximum is achieved at layered permutations.
Based on joint work with A. H. Morales, L. Petrov, D. Yeliussizov and D. Anderson, L. Petrov.”