Clusters, Coxeter-sortable elements and noncrossing partitions

By Nathan Reading. 2005

152 citations recorded by the source corpus.

We introduce Coxeter-sortable elements of a Coxeter group For finite we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in terms of their inversion sets and, in the classical cases, in terms of permutations

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

View Paper: https://www.semanticscholar.org/paper/6a9c4be40ac45f62f71887fdb5a4a904fcd32149

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Osamu Iyama
2. I. Reiten
3. B. Keller
4. Yuya Mizuno
5. Takuma Aihara
6. David E. Speyer
7. C. Amiot
8. G. Todorov

In Discrete Mathematics and Combinatorics

1. David E. Speyer
2. I. Reiten
3. Yuya Mizuno
4. Takuma Aihara
5. Hugh Thomas
6. Nathan Williams
7. Nathan Reading
8. Osamu Iyama

In Geometry and Topology

1. Osamu Iyama
2. I. Reiten
3. B. Keller
4. Yuya Mizuno
5. Takuma Aihara
6. David E. Speyer
7. Aslak Bakke Buan
8. G. Todorov

Citations in the corpus

• Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “4] and they show that elements of WQ satisfying a certain condition called being c-sortable [33] are in bijection with torsion classes ofmodK Q by this correspondence.”
• Noncrossing partitions and the shard intersection order (2009, unknown): “In [33], c-sortable elements are defined in terms of the combinatorics of reduced words, and in [34], the c-sortable elements are shown to be the bottom elements of congruence classes of Θc.”
• Classifying $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type (2013, background): “4] and they show that elements of WQ satisfying a certain condition called being c-sortable [33] are in bijection with torsion classes of modK Q by this correspondence.”
• A categorification of non-crossing partitions (2013, background): “Such Hom-free sets were studied by various authors [10, 23, 43, 44, 12].”
• Sortable elements and Cambrian lattices (2005, methodology): “(The rank-two subarrangements of A are exactly the collections of reflecting hyperplanes of the rank-two parabolic subgroups considered in [21].”
• Sortable elements in infinite Coxeter groups (2008, background): “All five of these results are given uniform proofs in this paper, so that in particular all results of [32, 33, 34] now have uniform proofs.”
• Rowmotion in slow motion (2017, background): “) It follows from Reading’s uniform bijection between c-sortable elements and c-noncrossing partitions [49] that x is determined by D(x) and also by U(x): D(x) encodes a factorization of a c-noncrossing partition π, and U(x) similarly encodes its Kreweras complement, the noncrossing partition Krewc(”
• Cambrian Lattices (2004, unknown): “In a subsequent paper [26], motivated by the study of Cambrian lattices, we define sortable elements of a Coxeter group W and use them to make a bijective connection between clusters and noncrossing partitions.”
• Cambrian fans (2006, methodology): “The c-Cambrian fan and the c-cluster fan The following simple lemma, which is [28, Lemma 8.5], is a key ingredient in the proof (in [28]) of Theorem 5.2 and in the results of this paper.”
• Preprojective algebras and c‐sortable words (2010, background): “c is unique for any c-sortable element of WQ (see [27]).”