From the Tamari lattice to Cambrian lattices and beyond

TL;DR

This paper explores the development of Cambrian lattices and fans from Tamari lattices, emphasizing sortable elements and their applications in Coxeter-Catalan combinatorics and cluster algebras.

Contribution

It introduces Cambrian lattices and fans within Coxeter groups and highlights sortable elements as a key combinatorial tool for their study.

Findings

Defined Cambrian lattices and fans for finite Coxeter groups.

Connected Cambrian structures to Coxeter-Catalan combinatorics.

Applied sortable elements to cluster algebra theory.

Abstract

In this chapter, we trace the path from the Tamari lattice, via lattice congruences of the weak order, to the definition of Cambrian lattices in the context of finite Coxeter groups, and onward to the construction of Cambrian fans. We then present sortable elements, the key combinatorial tool for studying Cambrian lattices and fans. The chapter concludes with a brief description of the applications of Cambrian lattices and sortable elements to Coxeter-Catalan combinatorics and to cluster algebras.