Jordan recoverability of some subcategories of modules over gentle algebras

TL;DR

This paper investigates the conditions under which modules over gentle algebras are uniquely determined by their generic Jordan form, focusing on specific subcategories of indecomposable representations.

Contribution

It provides a characterization of vertices in gentle quivers where modules are uniquely identified by their generic Jordan form.

Findings

Vertices with this property are explicitly characterized.

Modules over certain subcategories are recoverable from Jordan forms.

Results connect representation theory with linear algebraic invariants.

Abstract

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowro\'{n}ski in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.