Cokernels of random matrix products and flag Cohen--Lenstra heuristic

Abstract

In [NVP22], Nguyen and Van Peski raised the question of whether the surjective flag of ${\mathbb{Z}}_p$-modules modeled by $\mathrm{cok}(M_1\cdots M_k)\twoheadrightarrow \cdots \twoheadrightarrow \mathrm{cok}(M_1)$ for independent random matrices $M_1,\dots ,M_k\in {\mathrm{Mat}}_n({\mathbb{Z}}_p)$ satisfies the Cohen--Lenstra heuristic. We answer the question affirmatively when $M_1,\dots ,M_k$ follow the Haar measure, and our proof demonstrates how classical ideas in Cohen--Lenstra heuristic adapt naturally to the flag setting. We also prove an analogue for non-square matrices.

8 pages