From dual canonical bases to positroidal subdivisions

TL;DR

This paper explores the relationship between dual canonical bases of Grassmannian cluster algebras and positroidal subdivisions of hypersimplices, establishing new correspondences and conjectures for higher dimensions.

Contribution

It demonstrates that each tableau in the dual canonical basis induces a positroidal subdivision, and conjectures a formula for counting split subdivisions for all k.

Findings

Rectangular semi-standard Young tableaux induce positroidal subdivisions

For Gr(2,n), non-frozen prime tableaux correspond to coarsest subdivisions

Conjectural formula for counting split subdivisions

Abstract

The Grassmannian cluster algebra $\mathbb{C}[\text{Gr}(k, n)]$ admits a distinguished basis known as the dual canonical basis, whose elements correspond to rectangular semi-standard Young tableaux with $k$ rows and with entries in $[n]$. We establish that each such tableau induces a positroidal subdivision of the hypersimplex $\Delta(k,n)$ via a map introduced by Speyer and Williams. For $\text{Gr}(2,n)$, we prove that non-frozen prime tableaux correspond precisely to the coarsest positroidal subdivisions of $\Delta(2,n)$. Furthermore, we present computational evidence extending these results to $k>2$. In the process, we formulate a conjectural formula for the number of split positroidal subdivisions of $\Delta(k,n)$ for any $k \ge 2$ and explore the deep connections between the polyhedral combinatorics of $\Delta(k,n)$ and the dual canonical basis of $\mathbb{C}[\text{Gr}(k, n)]$.