Asymptotically maximal Schubitopes

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 ${\hat{I}}^2(n):={\max }_{w\in S_n}|\mathrm{supp}({\mathfrak{S}}_w)|$. 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 $β^{\mathfrak G}(n):=\max_{w\in S_n}|\mathrm{supp}(\mathfrak G_w)|$.

7 pages. v2: final version, to appear at Bull. Lond. Math. Soc