Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

Abstract

For an element w of a simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory of the module category over the preprojective algebra of Dynkin type. This paper studies categorical properties of using the root system. We show that simple objects in bijectively correspond to Bruhat inversion roots of w, and obtain a combinatorial criterion for to satisfy the Jordan-Hölder property (JHP). 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-Mélou and Butler in the study of Schubert varieties.