SIGMA 22 (2026), 016, 8 pages      arXiv:2509.16178      https://doi.org/10.3842/SIGMA.2026.016
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

Asymptotics for the Enumeration of Commuting Matrices over Finite Fields

Kathrin Bringmann ^a, Shane Chern ^b, Johann Franke ^a and Bernhard Heim ^a
a) University of Cologne, Department of Mathematics and Computer Science, Weyertal 86-90, 50931 Cologne, Germany
^b) Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Wien 1090, Austria

Received September 22, 2025, in final form January 19, 2026; Published online February 19, 2026

Abstract
We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.

Key words: asymptotic expressions; Cohen-Lenstra series; commuting matrices; finite fields; nilpotent classes.

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