A uniform bijection between nonnesting and noncrossing partitions

Armstrong, Drew; Stump, Christian et Thomas, Hugh (2013). « A uniform bijection between nonnesting and noncrossing partitions ». Transactions of the American Mathematical Society, 365(8), pp. 4121-4151.

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Résumé

In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we use Panyushev's map, together with the well-known Kreweras complement, to construct a bijection between nonnesting and noncrossing partitions. Our map is defined uniformly for all root systems, using a recursion in which the map is assumed to be defined already for all parabolic subsystems. Unfortunately, the proof that our map is well defined, and is a bijection, is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner.

Type: Article de revue scientifique
Mots-clés ou Sujets: Weyl groups, Coxeter groups, noncrossing partitions, nonnesting partitions,cyclic sieving phenomenon, bijective combinatorics.
Unité d'appartenance: Faculté des sciences > Département de mathématiques
Déposé par: Hugh R. Thomas
Date de dépôt: 18 mai 2016 19:48
Dernière modification: 30 mai 2016 20:07
Adresse URL : http://www.archipel.uqam.ca/id/eprint/8475

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