Asymptotically maximal Schubitopes
Jack Chen-An Chou, Linus Setiabrata
TL;DR
The paper investigates the asymptotics of the support sizes of Schubert and Grothendieck polynomials. It introduces the maximal support functions $\beta(n)$ and $\beta^{\mathfrak G}(n)$ and proves that their leading growth is governed by $\exp(n\ln n)$, with layered permutations achieving near-optimal bounds. A key technical tool is the Schubitope framework and a recursion for layered permutations that multiplies the support by factorial factors, yielding $|\mathrm{supp}(\mathfrak S_w)| \ge \prod_k \lfloor n/2^k\rfloor!$ for a carefully chosen block structure, and hence $\lim_{n\to\infty}\frac{\ln \beta(n)}{n\ln n}=1$ with explicit subleading terms. For Grothendieck polynomials, a fireworks-layered construction gives $|\mathrm{supp}(\mathfrak G_w)| \ge n!/n^{\sqrt{2n}+1}$, leading to $\lim_{n\to\infty}\frac{\ln(\beta^{\mathfrak G}(n)) - n\ln n}{n} = -1$, i.e., the Grothendieck case matches the Schubert case up to subexponential factors. These results connect to Schubitopes and address questions posed by Guo–Lin about maximal supports.
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 $β(n):= \max_{w\in S_n}|\mathrm{supp}(\mathfrak S_w)|$. 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 $β^{\mathfrak G}(n):=\max_{w\in S_n}|\mathrm{supp}(\mathfrak G_w)|$.