Commuting matrices via commuting endomorphisms

TL;DR

This paper develops a general framework to relate counting problems of commuting matrices to endomorphisms on finite abelian p-groups, providing new insights into module counting over nonreduced curves and linking classical problems to a conjecture by Onn.

Contribution

It introduces a unified approach to reduce matrix counting problems to endomorphism problems on finite abelian p-groups, connecting various complex counting problems.

Findings

Reduced some matrix counting problems to endomorphism problems.

Counted finite modules on nonreduced curves over finite fields.

Connected classical commuting matrix problems to Onn's conjecture.

Abstract

Evidences have suggested that counting representations are sometimes tractable even when the corresponding classification problem is almost impossible, or "wild" in a precise sense. Such counting problems are directly related to matrix counting problems, many of which are under active research. Using a general framework we formulate for such counting problems, we reduce some counting problems about commuting matries to problems about endomorphisms on all finite abelian $p$-groups. As an application, we count finite modules on some first examples of nonreduced curves over $\mathbb{F}_q$. We also relate some classical and hard problems regarding commuting triples of matrices to a conjecture of Onn on counting conjugacy classes of the automorphism group of an arbitrary finite abelian $p$-group.