Semibricks, torsion-free classes and the Jordan-Hölder property

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-Hölder property (WJHP) and Jordan-Hölder 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 $A$.

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