Mutually annihilating matrices, and a Cohen–Lenstra series for the nodal singularity

Abstract

We give a generating function for the number of pairs of n×n matrices (A,B) over a finite field that are mutually annihilating, namely, AB=BA=0. This generating function can be viewed as a singular analogue of a series considered by Cohen and Lenstra. We show that this generating function has a factorization that allows it to be meromorphically extended to the entire complex plane. We also use it to count pairs of mutually annihilating nilpotent matrices. This work is essentially a study of the motivic aspects about the variety of modules over C[u,v]/(uv) as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities.