## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

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.

Zoom meeting: https://us02web.zoom.us/j/85273867663?pwd=VjcyWHl6dWhLZUJId3dwbFRUQVVuUT09

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.