# Algebra Seminar Wednesday, November 25, 2020 — 2:30 PM EST

Yifeng Huang, University of Michigan - Ann Arbor

"A generating function for counting mutually annihilating matrices over a finite field"

In 1958, Fine and Herstein proved that the an n by n matrix over the finite field F_q has a probability of q^{-n} to be nilpotent. A clever application of this result can lead to the formula \sum_{H: abelian p-group} 1/|Aut(H)| = 1/((1-p^-1)(1-p^-2)...), which is fundamental in building the Cohen--Lenstra distribution of abelian p-groups. There are other matrix enumeration results, including the counting of pairs of commuting matrices (Feit and Fine) and the counting of pairs of commuting nilpotent matrices (Fulman), all presented as generating functions that can be expressed as infinite products of rational functions. I will explain why all these above are the special cases of one general problem related to a moduli space in algebraic geometry, and why the following is the next unknown case of the problem: count the number of pairs of n by n matrices (A,B) such that AB=BA=0 (hence the word "mutually annihilating" in the title). In my recent work, I gave a generating function that answers this question, and factorized it into the form 1/((1-x)(1-q^-1 x)(1-q^-2 x)...)^2 H(x), where H(x) is an entire holomorphic function given explicitly by an infinite sum. Interesting analytic properties of H(x) will be discussed.

### January 2023

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