Asymptotically maximal Schubitopes

TL;DR

This paper determines the asymptotic growth rates of the support sizes of Schubert and Grothendieck polynomials for specific layered permutations, revealing exponential and super-exponential behaviors.

Contribution

It identifies particular layered permutations with support sizes growing at specific asymptotic rates, providing new bounds for the maximal support sizes of these polynomials.

Findings

Support size of Schubert polynomials grows at least as fast as n!/4^n.

Support size of Grothendieck polynomials grows at least as fast as n!/e^{\sqrt{2n} \\ln(n)}.

Provides precise asymptotics for the growth rates of maximal support sizes.

Abstract

We find a layered permutation $w\in S_n$ whose Schubert polynomial $\mathfrak S_w(x_1, \dots, x_n)$ has support of size asymptotically at least $n!/4^n$. This gives precise asymptotics for the growth rate of $\beta(n):= \max_{w\in S_n}|\mathrm{supp}(\mathfrak S_w)|$. We find a different layered permutation $w\in S_n$ whose Grothendieck polynomial has support of size asymptotically at least $n!/e^{\sqrt{2n} \cdot \ln(n)}$ and obtain more precise asymptotics for the growth rate of $\beta^{\mathfrak G}(n):=\max_{w\in S_n}|\mathrm{supp}(\mathfrak G_w)|$.