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.
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.
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Keywords: Finite Coxeter groups, Noncrossing partitions, Bruhat order, cluster complex
Mot clés : groupes de Coxeter finis, partitions noncroisées, ordre de Bruhat, complexe d’amas
Biane, Philippe 1; Josuat-Vergès, Matthieu 1
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.3294/} }
TY - JOUR AU - Biane, Philippe AU - Josuat-Vergès, Matthieu TI - Noncrossing partitions, Bruhat order and the cluster complex JO - Annales de l'Institut Fourier PY - 2019 SP - 2241 EP - 2289 VL - 69 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3294/ DO - 10.5802/aif.3294 LA - en ID - AIF_2019__69_5_2241_0 ER -
%0 Journal Article %A Biane, Philippe %A Josuat-Vergès, Matthieu %T Noncrossing partitions, Bruhat order and the cluster complex %J Annales de l'Institut Fourier %D 2019 %P 2241-2289 %V 69 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3294/ %R 10.5802/aif.3294 %G en %F AIF_2019__69_5_2241_0
Biane, Philippe; Josuat-Vergès, Matthieu. Noncrossing partitions, Bruhat order and the cluster complex. Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2241-2289. doi : 10.5802/aif.3294. https://aif.centre-mersenne.org/articles/10.5802/aif.3294/
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