Splitting Subspaces of Linear Operators over Finite Fields

2020/12/15 by Divya Aggarwal, Aggarwal, Divya, Samrith Ram +1
Computer Science · Mathematics · #Coding theory and cryptography #Finite Group Theory Research #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2012.08411

Abstract

Let V be a vector space of dimension N over the finite field \mathbbF_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⊕ TW⊕ ⋯ ⊕ T^d-1W where d=N/m. Let σ(m,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. Denote by σ_q(m,d;τ) the number of m-dimensional splitting subspaces for a linear operator of similarity class type τ over an mathbbF_q-vector space of dimension md. For fixed values of m,d and τ, we show that σ_q(m,d;τ) is a polynomial in q.

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