Classifying τ-tilting modules over preprojective algebras of Dynkin type

We study support $τ$-tilting modules over preprojective algebras of Dynkin type. We classify basic support $τ$-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and other important objects in representation theory. We also study $g$-matrices of support $τ$-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of $g$-matrices and prove that cones given by $g$-matrices coincide with chambers of the associated root systems.

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