Based on paper:
In this talk, I will talk about the Jordan-Hölder property (JHP) for exact categories, which is a natural generalization of the uniqueness property of decompositions of modules into simples. First, I introduce a new invariant of exact categories, the Grothendieck monoids, show that (JHP) is equivalent to the free-ness of this monoid, and give a convenient numerical criterion for this.
Second, I will apply them to representation theory of artin algebras. Under a mild assumption, (JHP) is equivalent to that the number of projectives is equal to that of simples. For torsion-free classes of type A quiver, simple objects are described in terms of the combinatorics of the symmetric group: Bruhat inversions of c-sortable elements. Thus we can check the validity of (JHP) in a purely combinatorial way in this case.