The Catalan combinatorics of the hereditary artin algebras

TL;DR

This survey explores the categorification of non-crossing partitions via hereditary artin algebras, connecting combinatorics, representation theory, and Weyl group exponents, with a focus on Dynkin type A.

Contribution

It introduces a refined tilting theory framework using torsion triples and establishes bijections between modules and subcategories, linking combinatorics with algebraic structures.

Findings

Dynkin functions relate to Weyl group exponents

Identification of P with non-crossing partitions in type A

Connections between maximal chains, trees, and parking functions

Abstract

This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories and there is a wealth of bijections, the Ingalls-Thomas bijections, between sets of modules and subcategories. If H is representation-finite, one may look at the corresponding numbers of modules or subcategories. Such Dynkin functions (they attach to a Dynkin diagram an integer) are displayed in chapter 1. In a mysterious way, many Dynkin functions can be described using the exponents of the Weyl group. According to Shapiro and Kostant, the exponents are given by the height partition of the root poset. A recent result of Abe-Barakat-Cuntz-Hoge-Terao allows to determine them inductively, going up in a chain of ideals in the root poset, looking at the corresponding hyperplane arrangements. Chapter 4 deals with the case of the linearly oriented quiver of Dynkin type A. Here P is identified with the lattice NC of non-crossing partitions as introduced by Kreweras (now an important tool in several parts of mathematics, for example in free probability theory). We review some classical problems which are related to the maximal chains in NC: to count labeled trees as well as parking functions. The combinatorics of the Dynkin case A is just the combinatorics of the Catalan numbers; in an appendix, we discuss the nature of classical Catalan combinatorics.