From dual canonical bases to positroidal 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 $Δ(k,n)$ 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 $Δ(2,n)$. 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 $Δ(k,n)$ for any $k \ge 2$ and explore the deep connections between the polyhedral combinatorics of $Δ(k,n)$ and the dual canonical basis of $\mathbb{C}[\text{Gr}(k, n)]$.