Noncrossing partitions, Bruhat order and the cluster complex
[Partitions non-croisées, ordre de Bruhat et complexe d’amas]
Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 2241-2289.

Nous introduisons deux relations d’ordre sur les groupes de Coxeter finis qui raffinent l’ordre absolu et l’ordre de Bruhat, et obtenons quelques propriétés essentielles. En particulier, nous étudions la restriction de ces ordres aux partitions non-croisées, et montrons que les intervalles pour ces ordres peuvent être comptés en termes du complexe d’amas. Les propriétés de nos ordres permettent de revoir divers résultats en combinatoire des groupes de Coxeter finis, tels que les triangles de Chapoton et leurs relations, l’énumération des réflexions à support pleins, les bijections entre partitions non-croisées et amas.

We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular, we study the restriction of these orders to noncrossing partitions and show that the intervals for these orders can be enumerated in terms of the cluster complex. The properties of our orders permit to revisit several results in Coxeter combinatorics, such as the Chapoton triangles and how they are related, the enumeration of reflections with full support, the bijections between clusters and noncrossing partitions.

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DOI : https://doi.org/10.5802/aif.3294
Classification : 05A15,  20F55
Mots clés : groupes de Coxeter finis, partitions noncroisées, ordre de Bruhat, complexe d’amas
@article{AIF_2019__69_5_2241_0,
     author = {Biane, Philippe and Josuat-Verg\`es, Matthieu},
     title = {Noncrossing partitions, Bruhat order and the cluster complex},
     journal = {Annales de l'Institut Fourier},
     pages = {2241--2289},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {5},
     year = {2019},
     doi = {10.5802/aif.3294},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3294/}
}
Biane, Philippe; Josuat-Vergès, Matthieu. Noncrossing partitions, Bruhat order and the cluster complex. Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 2241-2289. doi : 10.5802/aif.3294. http://archive.numdam.org/articles/10.5802/aif.3294/

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