Sortable elements and Cambrian lattices

By Nathan Reading. 2005

104 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bf0d5fae4fe98b632dd949a75062743d08a7a882

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Citations in the corpus

Powers of Coxeter elements in infinite groups are reduced (2007, background): “in [5]; it is the opposite of the one used in [10] and [11].”
Noncrossing partitions and the shard intersection order (2009, unknown): “The set πc ↓ (W) can be characterized combinatorially [34] as the set of c-sortable elements of W .”
Rowmotion in slow motion (2017, background): “This lattice can also be realized as a quotient of weak order via the projection π ↓ : W → Sort(W, c) that sends an element in W to the largest c-sortable element below it [50].”
Sortable elements in infinite Coxeter groups (2008, background): “This section is included for the benefit of those citing specific results from previous papers in the series [32, 33, 34].”
Permutahedra and generalized associahedra (2007, background): “Reading shows that the c-sortable elements constitute a sublattice of the weak order of W which is called the c-Cambrian lattice, [18, 20].”
Subword complexes, cluster complexes, and generalized multi-associahedra (2011, background): “of cluster complexes can be extended to all finite Coxeter groups [48, 49].”
Cambrian fans (2006, methodology): “The results of this section rely (indirectly through results proved or quoted in Section 3) on nontrivial results from [28] and [29].”
Clusters, Coxeter-sortable elements and noncrossing partitions (2005, background): “Using results from [20] and [ 22 ], one associates a complete fan to c-sortable elements.”
Brick polytopes of spherical subword complexes and generalized associahedra (2011, methodology): “eading and D. Speyer in [Rea06,Rea07b,RS09]. We rst describe in detail the bijective connections between the facets of the subword complex SC(cw (c)) and dierent Coxeter-Catalan families studied in [Rea07b]: c-clusters, c-sortable elements, and c-noncrossing partitions and subspaces. In particular, ”