Splitting Subspaces of Linear Operators over Finite Fields

Let $V$ be a vector space of dimension $N$ over the finite field $\mathbb{F}_q$ 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 $V=W\oplus TW\oplus \cdots \oplus T^{d-1}W$ 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 $\\mathbb{F}_q$-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 $q$.