Suborbits of subspaces of type (m, k) under finite singular general linear groups

Suppose F(q)(n+l) denotes the (n + l)-dimensional vector space over a finite field F(q) and GL(n+l,n)(F(q)) denotes the corresponding singular general linear group. All the subspaces of type (m, k) form an orbit under GL(n+l,n)(Fq), denoted by M(m,k: n + l, n). Let A be the set of all the orbitals of (GL(n+l,n)(F(q)), M(m, k: n + l, n)). Then (m (m, k: n + l, n), A) is a symmetric association scheme. In this paper, we determine all the orbitals and the rank of (GL(n+1,n)(F(q)), M(m, k: n + l, n)), calculate the length of each suborbit. Finally, we compute all the intersection numbers of the symmetric association scheme (m (m, k: n + l, n), A), where k = 1 or k = l - 1. (C) 2009 Elsevier Inc. All rights reserved.