Profile vectors in the lattice of subspaces

The profile vector f(U)∈Rn+1f(U)∈Rn+1 of a family UU of subspaces of an nn-dimensional vector space VV over GF(q)GF(q) is a vector of which the iith coordinate is the number of subspaces of dimension ii in the family UU(i=0,1,…,n)(i=0,1,…,n). In this paper, we determine the profile polytope of intersecting families (the convex hull of the profile vectors of all intersecting families of subspaces).