Torsion pairs for quivers and the Weyl groups

We give an interpretation of the map π ^c defined by Reading, which is a map from the elements of a Coxeter group to the c -sortable elements, in terms of the representation theory of preprojective algebras. Moreover, we study a close relationship between c -sortable elements and torsion pairs, and give an explicit description of the cofinite torsion classes in the context of the Coxeter group. As a consequence, we give a proof of some conjectures proposed by Oppermann, Reiten, and the second author.