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

2021/10/29 by Huang, Yifeng · 4 citations
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2110.15566

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 \mathbb C[u,v]/(uv) as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities.

Cited by

• Matrix Points on Varieties
• Asymptotics for the enumeration of commuting matrices over finite fields
• Coh zeta functions for inert quadratic orders
• On matrices commuting with their Frobenius

Related

• Cohen Lenstra Partitions and Mutually Annihilating Matrices over a\n Finite Field
• Varieties of pairs of nilpotent matrices annihilating each other
• On Numbers of Tuples of Nilpotent Matrices over Finite Fields under Simultaneous Conjugation
• Counting on the variety of modules over the quantum plane
• Jordan forms for mutually annihilating nilpotent pairs