Association schemes on the Schubert cells of a Grassmannian

Let $\mathbb{F}$ be any field. The Grassmannian $\mathrm{Gr}(m,n)$ is the set of $m$-dimensional subspaces in $\mathbb{F}^n$, and the general linear group $\mathrm{GL}_n(\mathbb{F})$ acts transitively on it. The Schubert cells of $\mathrm{Gr}(m,n)$ are the orbits of the Borel subgroup $\mathcal{B} \subset \mathrm{GL}_n(\mathbb{F})$ on $\mathrm{Gr}(m,n)$. We consider the association scheme on each Schubert cell defined by the $\mathcal{B}$-action and show it is symmetric and it is the generalized wreath product of one-class association schemes, which was introduced by R. A. Bailey [European Journal of Combinatorics 27 (2006) 428--435].