Splitting subspaces of linear operators over finite fields

Let V be a vector space of dimension N over the finite field F-q and T be a linear operator on V. Given an integer m that divides N, an m-dimensional subspace W of V is T -splitting if V = W (R) TW (R) middot middot middot (R) Td-1W where d = N/m. Let sigma(m, d; T) denote the number of m-dimensional T-splitting subspaces. Determining sigma(m, d;T) for an arbitrary operator T is an open problem. We prove that sigma(m, d; T) depends only on the similarity class type of T and give an explicit formula in the special case where T is cyclic and nilpotent. Denote by sigma(q)(m, d; Tau) the number of m-dimensional splitting subspaces for a linear operator of similarity class type Tau over an F-q-vector space of dimension md. For fixed values of m, d and Tau, we show that sigma(q)(m, d; Tau) is a polynomial in q. (C) 2021 Elsevier Inc. All rights reserved.