Rook theory of the finite general linear group

TL;DR

This paper develops a $q$-analogue of rook theory for matrices over finite fields with fixed rank and support, establishing identities, connections to coding theory, and addressing conjectures.

Contribution

It introduces a new $q$-rook theory framework, defines $q$-hit numbers, and proves identities and conjecture results, expanding the combinatorial and algebraic understanding.

Findings

Established identities for $q$-hit and $q$-rook numbers

Connected $q$-rook theory with coding theory techniques

Resolved a polynomiality conjecture and provided a counterexample to a positivity conjecture

Abstract

Matrices over a finite field having fixed rank and restricted support are a natural $q$-analogue of rook placements on a board. We develop this $q$-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding theory, we show that these $q$-hit and $q$-rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier $q$-analogues of rook theory, as well as settling a polynomiality conjecture and finding a counterexample of a positivity conjecture of the authors and Klein.