Splitting Subspaces of Linear Operators over Finite Fields

TL;DR

This paper investigates the enumeration of splitting subspaces of linear operators over finite fields, providing explicit formulas for certain classes and showing polynomial dependence on the field size.

Contribution

It proves that the count of splitting subspaces depends only on the similarity class type of the operator and derives explicit formulas for cyclic nilpotent operators.

Findings

Number of splitting subspaces depends only on similarity class type.

Explicit formula derived for cyclic nilpotent operators.

Counting function is polynomial in the size of the finite field.

Abstract

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 $q$.