Let (Formula presented.) be any field. The Grassmannian (Formula presented.) is the set of m-dimensional subspaces in (Formula presented.), and the general linear group (Formula presented.) acts transitively on it. The Schubert cells of (Formula presented.) are the orbits of the Borel subgroup (Formula presented.) on (Formula presented.). We consider the association scheme on each Schubert cell defined by the (Formula presented.)-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).