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Sortable elements and Cambrian lattices. (English) Zbl 1184.20038
Summary: 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 $$W$$.

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 06A07 Combinatorics of partially ordered sets 06B10 Lattice ideals, congruence relations 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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