Enumeration of subspaces by dimension sequence

Abstract

Let F be the finite field with q elements and let V be an m-dimensional vector space over F. Fix a linear endomorphism A of V. Suppose that X ⊆ V is a subspace. Let X (0) = 0, X (1) = X, and X ( k) = X ( k − 1) + A k − 1 ( X) for k > 1. For k ⩾ 1 let j k be the dimension of the quotient space X (k) X (k−1) . The m-tuple j = ( j 1, j 2, …, j m ) is called the dimension sequence of the subspace X. In this paper we consider the problem of determining the numbers C(j) of subspaces of V which have dimension sequence j. We derive two surprisingly simple formulas. First, if A is a shift operator (nilpotent with 1-dimensional null space) then C(j)= ∏ k=2 m q j 2 k j k−1 j k . Second, if A is simple (no non-trivial invariant F-subspaces) then C(j)= q m−1 q j1−1 ∏ k=2 m q jk(jk−1) j k−1 j k . We known of no simple counting proof of these formulas. Our derivation finds the C(j)'s as the solution to a set of linear equations obtained with Möbius inversion on the lattice of subspaces of V.