Enumerating partial linear transformations in a similarity class
Akansha Arora, Samrith Ram
TL;DR
The paper addresses counting the sizes of similarity classes of linear transformations defined on subspaces of a finite-dimensional vector space over ${\mathbb F}_q$. It introduces a complete set of similarity invariants for maps $T\in L(W,V)$ via a defect-partition $\lambda$ derived from a chain of $T$-invariant subspaces and an ordered set of invariant factors $\mathcal I$ of the restriction to the maximal invariant subspace, showing that $(\lambda, \mathcal I)$ classify similarity. It derives explicit formulas for the sizes $|\mathcal C(\lambda,\mathcal I)|$ and specialized counts for simple maps $|\mathcal C(\lambda,\emptyset)|$ and their domain-fixed variants, recovering Hall’s classical results in the full-space case and generalizing them to subspace domains. By decomposing arbitrary maps into operator and simple parts and counting lifts, it provides a unified enumeration framework in terms of $q$-binomial coefficients and the invariant-data $(\lambda,\mathcal I)$, with connections to control theory via zero-kernel pairs.
Abstract
Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to be similar if there exists a linear isomorphism $S:V\to V$ with $SW=\widetilde{W}$ such that $S\circ T=\widetilde{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.