Enumerating partial linear transformations in a similarity class

Abstract

Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb{F}}_q$ and suppose $W$ and $\tilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\tilde{T}:\tilde{W}\to V$ are said to be similar if there exists a linear isomorphism $S:V\to V$ with $SW=\tilde{W}$ such that $S\circ T=\tilde{T}\circ S$. Given a linear map $T$ defined on a subspace $W$ of $V$, we give an explicit formula for the number of linear maps that are similar to $T$. Our results extend a theorem of Philip Hall that settles the case $W=V$ where the above problem is equivalent to counting the number of square matrices over ${\mathbb{F}}_q$ in a conjugacy class.

15 pages, 3 figures. One minor typo corrected in the main result