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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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