Counting on the variety of modules over the quantum plane

TL;DR

This paper derives a generating function to count pairs of matrices over finite fields satisfying a quantum commutation relation, generalizing known results for commuting matrices and connecting to the geometry of modules over the quantum plane.

Contribution

It provides a new generating function for counting matrix pairs satisfying a quantum relation, extending classical results and linking algebraic combinatorics with algebraic geometry.

Findings

Derived a generating function for pairs of matrices satisfying AB=ζBA.

Generalized Feit and Fine's counting of commuting matrices.

Connected matrix counting to the geometry of modules over the quantum plane.

Abstract

Let $\zeta$ be a fixed nonzero element in a finite field $\mathbb F_q$ with $q$ elements. In this article, we count the number of pairs $(A,B)$ of $n\times n$ matrices over $\mathbb F_q$ satisfying $AB=\zeta BA$ by giving a generating function. This generalizes a generating function of Feit and Fine that counts pairs of commuting matrices. Our result can be also viewed as the point count of the variety of modules over the quantum plane $xy=\zeta yx$, whose geometry was described by Chen and Lu.