  For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott
defined a subcategory $\mathcal{F}(w)$ of a module category over a
preprojective algebra of Dynkin type. This paper aims at studying categorical
properties of $\mathcal{F}(w)$ via its connection with the root system. We show
that by taking dimension vectors, simple objects in $\mathcal{F}(w)$
bijectively correspond to Bruhat inversion roots of $w$. As an application, we
obtain a combinatorial criterion for $\mathcal{F}(w)$ to satisfy the
Jordan-H\"older property (JHP). To achieve this, we develop a method to find
simple objects in a general torsion-free class by using a brick sequence
associated to a maximal green sequence of it. For type A case, we give a
diagrammatic construction of simple objects, and show that (JHP) can be
characterized via a forest-like permutation, introduced by Bousquet-M\'elou and
Butler in the study of Schubert varieties.
