The Jordan-H\"older property and Grothendieck monoids of exact categories

TL;DR

This paper characterizes when the Jordan-H"older property holds in exact categories using Grothendieck monoids and applies these results to the representation theory of artin algebras, with combinatorial insights from symmetric groups.

Contribution

It provides a new criterion for the Jordan-H"older property in exact categories based on Grothendieck monoids and applies this to classify torsion-free classes in type A quivers.

Findings

JHP holds iff the Grothendieck monoid is free.

JHP in torsion classes iff the number of indecomposables equals simples.

Combinatorial criterion for JHP using Bruhat inversions.

Abstract

We investigate the Jordan-H\"older property (JHP) in exact categories. First, we show that (JHP) holds in an exact category if and only if the Grothendieck monoid introduced by Berenstein and Greenstein is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next, we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of indecomposable projectives is equal to that of simples. We study torsion-free classes in a quiver of type A in detail using the combinatorics of symmetric groups. We introduce Bruhat inversions of permutations and show that simples in a torsion-free class are in bijection with Bruhat inversions of the corresponding $c$-sortable element. We use this to give a combinatorial criterion for (JHP).