Association schemes on the Schubert cells of a Grassmannian

TL;DR

This paper studies association schemes on Schubert cells of Grassmannians, showing they are symmetric and can be expressed as generalized wreath products of one-class schemes, revealing their algebraic structure.

Contribution

It introduces a new perspective by characterizing association schemes on Schubert cells as generalized wreath products, extending prior combinatorial frameworks.

Findings

Association schemes on Schubert cells are symmetric.

These schemes are generalized wreath products of one-class schemes.

The structure generalizes previous combinatorial models.

Abstract

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].