Sortable elements and Cambrian lattices

Abstract

We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify this congruence as the Cambrian congruence on W, so that the Cambrian lattice is the weak order on Coxeter-sortable elements. These results exhibit W-Catalan combinatorics arising in the context of the lattice theory of the weak order on W.