Based on paper:
For a Dynkin quiver, Gabriel’s theorem gives a bijection between indecomposable representations of it and positive roots of the corresponding root system. Using this, every element of the Weyl group gives a subcategory of the module category by taking the inversion set. Under this construction, it was shown by Ingalls-Thomas that ”good” elements in the Weyl group are in bijection with ”good” subcategories of the module category, namely, torsion-free classes.
In this talk, I will restrict our attention to type A case (where everything is explicit), and give an overview of these results. Then I will explain my recent result (arXiv:1908.05446), which gives a bijection between simple objects and Bruhat inversions, and characterize whether a torsion-free class satisfies the Jordan-Holder property or not in a purely combinatorial way.