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

Yifeng Huang

Abstract

In [NVP22], Nguyen and Van Peski raised the question of whether the surjective flag of $\mathbb Z_p$-modules modeled by $\mathrm{cok}(M_1\cdots M_k)\twoheadrightarrow \dots\twoheadrightarrow \mathrm{cok}(M_1)$ 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.

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 -modules modeled by for independent random matrices satisfies the Cohen--Lenstra heuristic. We answer the question affirmatively when 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.
Paper Structure (11 sections, 9 theorems, 32 equations)

This paper contains 11 sections, 9 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Background
  4. Our result
  5. Further applications
  6. Preliminaries
  7. Dictionary of flags
  8. Proof of Theorem \ref{['thm:main']}
  9. Proof of Theorem \ref{['thm:nonsquare']}
  10. Probability measures on flags
  11. Relation to matrices over finite fields

Key Result

Theorem 1.1

Fix $k\in {\mathbb Z}_{\geq 1}$. Let $M_1,\dots,M_k\in \mathop{\mathrm{Mat}}\nolimits_n({\mathbb Z}_p)$ be independent and Haar-random, and fix $\mathbf{G}=(G_k\twoheadrightarrow \dots\twoheadrightarrow G_1)\in \mathbf{Fl}_k$ such that $\lvert G_k\rvert<\infty$. Then for $n\geq r(G_k)$, In particular, when $n\to\infty$,

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Definition 1.3
  • Corollary 1.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 12 more