Splitting Subspaces of Linear Operators over Finite Fields

Abstract

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, d; T ) denote the number of m-dimensional T -splitting subspaces. Determining σ(m, d; T ) for an arbitrary operator T is an open problem. We prove that σ(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. We also show that σ(m, d; T ) is a polynomial in q. Contents 1. Introduction 1 2. Counting Flags of Subspaces 3 3. Similarity Class Type and Splitting Subspaces 6 4. Cyclic Nilpotent Operators 8 5. Polynomiality of σ(m, d; T ) 12 6. Acknowledgements 12 References 12