The Jordan-Hölder property, Grothendieck monoids and Bruhat inversions

Date:

Seminar Talk in Ring Theory and Representation Theory Seminar at Nagoya University, Japan

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Abstract

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.

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