Cataland: Why the Fuss?

TL;DR

This paper introduces a unified framework for Fuss-Catalan combinatorics related to Coxeter systems, connecting noncrossing partitions, clusters, and sortable elements through new theoretical insights and representation theory interpretations.

Contribution

It extends Fuss-Catalan generalizations to sortable elements by lifting the theory to positive Artin monoids, unifying three key combinatorial objects.

Findings

Unified framework for noncrossing Fuss-Catalan combinatorics

New interpretation of sortable elements in Artin monoids

Connections to representation theory of hereditary Artin algebras

Abstract

The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.