Semibricks, torsion-free classes and the Jordan-H\"{o}lder property

TL;DR

This paper introduces the weak and strong Jordan-Hölder properties in extriangulated categories, characterizes when certain subcategories satisfy these properties, and provides combinatorial criteria for torsion-free classes in type A quiver representations.

Contribution

It defines the weak and Jordan-Hölder properties in extriangulated categories and characterizes when filtration subcategories satisfy these properties, linking to combinatorial criteria in quiver representations.

Findings

$ T$ satisfies (WJHP) in $ ext{extriangulated}$ categories.

$ T$ satisfies (JHP) iff $ X$ is proper.

Provides combinatorial criteria for (JHP) in type A quiver categories.

Abstract

Let $\mathscr{C}$ be an extriangulated category and $\mathcal{X}$ be a semibrick in $\mathscr{C}$. Let $\mathcal{T}$ be the filtration subcategory generated by $\mathcal{X}$. We introduce the weak Jordan-H\"{o}lder property (WJHP) and Jordan-H\"{o}lder property (JHP) in $\mathscr{C}$ and show that $\mathcal{T}$ satisfies (WJHP). Furthermore, $\mathcal{T}$ satisfies (JHP) if and only if $\mathcal{X}$ is proper. Using reflection functors and $c$-sortable elements, we give a combinatorial criterion for the torsion-free class satisfying (JHP) in the representation category of a quiver of type $A$.