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Sortable elements and Cambrian lattices

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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.

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Received December 22, 2005; accepted in final form March 19, 2006.

The author was partially supported by NSF grants DMS-0202430 and DMS-0502170.

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Reading, N. Sortable elements and Cambrian lattices. Algebra univers. 56, 411–437 (2007).

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2000 Mathematics Subject Classification: