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

Abstract

We give a generating function for the number of pairs of $n\times n$ 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 $\mathbb{C}[u,v]/(uv)$ as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities.

24 pages

References in corpus (2)

• Pairs of mutually annihilating operators
• Varieties of modules over the quantum plane