Counting on the variety of modules over the quantum plane

Abstract

Let ζ be a fixed nonzero element in a finite field Fq with q elements. In this article, we count the number of pairs (A,B) of n × n matrices over Fq satisfying AB = ζBA 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 = ζyx, whose geometry was described by Chen and Lu.