Association Schemes on the Schubert Cells of a Grassmannian

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 Bailey (Eur J Comb 27(3):428–435, 2006).