Canonically Jordan recoverable categories for modules over the path algebra of $A_n$ type quivers

TL;DR

This paper characterizes subcategories of representations over $A_n$ quivers where the generic Jordan form data uniquely determines the module, providing a combinatorial description of these canonically Jordan recoverable categories.

Contribution

It offers a combinatorial characterization of canonically Jordan recoverable subcategories for modules over $A_n$ quivers, enabling module recovery from Jordan form data.

Findings

Identifies which subcategories are canonically Jordan recoverable.

Provides a combinatorial criterion for recoverability.

Establishes an inversion procedure for the Jordan form data.

Abstract

Let $Q$ be a quiver of $A_n$ type and $\mathbb{K}$ be an algebraically closed field. A nilpotent endomorphism of a quiver representation induces a linear transformation of the vector space at each vertex. Generically among all nilpotent endomorphisms of a fixed representation $X$, there exists a well-defined Jordan form of each of these linear transformations $\operatorname{GenJF}(X)$, called the generic Jordan form data of $X$. A subcategory of $\operatorname{rep}(Q)$ is Jordan recoverable if we can recover $X$ up to isomorphism from its generic Jordan form data. There is a procedure which allows one to invert the map from representations to generic Jordan form data. The subcategories for which this procedure works are called canonically Jordan recoverable. We focus on the subcategories of $\operatorname{rep}(Q)$ that are canonically Jordan recoverable, and we give a combinatorial characterization of them.