Schubert unions in Grassmann varieties

We study subsets of Grassmann varieties G(l,m) over a field F, such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study unions of Schubert cycles of Grassmann varieties G(l,m) over a field F. We compute their linear span and, in positive characteristic, their number of F_q-rational points. Moreover, we study a geometric duality of such unions, and give a combinatorial interpretation of this duality. We discuss the maximum number of F_q-rational points for Schubert unions of a given spanning dimension, and as an application to coding theory, we study the parameters and support weights of the well-known Grassmann codes. Moreover, we determine the maximum Krull dimension of components in the intersection of G(l,m) and a linear space of given dimension in the Plücker space.