no. 5
Noncrossing partitions, Bruhat order and the cluster complex
Biane, Philippe1; Josuat-Vergès, Matthieu1
1 Laboratoire d’Informatique Gaspard Monge (LIGM), Université Paris-Est Marne-la-Vallée (UPEM), CNRS Marne-la-Vallée (France)
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2241-2289.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3294
Classification: 05A15, 20F55
Keywords: Finite Coxeter groups, Noncrossing partitions, Bruhat order, cluster complex
Biane, Philippe 1; Josuat-Vergès, Matthieu 1

1 Laboratoire d’Informatique Gaspard Monge (LIGM), Université Paris-Est Marne-la-Vallée (UPEM), CNRS Marne-la-Vallée (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@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/
UR  - https://doi.org/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://doi.org/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/

[1] Armstrong, Drew Generalized noncrossing partitions and combinatorics of Coxeter groups, Memoirs of the American Mathematical Society, 202, American Mathematical Society, 2009 no. 949, x+159 pages | MR | Zbl

[2] Athanasiadis, Christos A. On some enumerative aspects of generalized associahedra, Eur. J. Comb., Volume 28 (2007), pp. 1208-1215 | DOI | MR | Zbl

[3] Athanasiadis, Christos A.; Brady, Thomas; McCammond, Jon; Watt, Colum h-vectors of generalized associahedra and noncrossing partitions, Int. Math. Res. Not., Volume 2006 (2006) no. 12, 69705, 28 pages | MR | Zbl

[4] Athanasiadis, Christos A.; Savvidou, Christina The local h-vector of the cluster subdivision of a simplex, Sémin. Lothar. Comb., Volume 66 (2012), B66c, 21 pages | MR | Zbl

[5] Barnard, Emily; Reading, Nathan Coxeter-biCatalan combinatorics, J. Algebr. Comb., Volume 47 (2018) no. 2, pp. 241-300 | DOI | MR | Zbl

[6] Belinschi, Serban T.; Nica, Alexandru η-series and and a boolean Bercovici-Pata bijection for bounded k-tuples, Adv. Math., Volume 217 (2008), pp. 1-41 | DOI | MR | Zbl

[7] Bessis, David The dual braid monoid, Ann. Sci. Éc. Norm. Supér., Volume 36 (2003) no. 5, pp. 647-683 | DOI | MR | Zbl

[8] Biane, Philippe Some properties of crossings and partitions, Discrete Math., Volume 175 (1997) no. 1-3, pp. 41-53 | DOI | MR | Zbl

[9] Biane, Philippe; Josuat-Vergès, Matthieu Minimal factorizations of a cycle: a multivariate generating function, FPSAC 2016 (Vancouver, 2016) (Discrete Mathematics and Theoretical Computer Science) (2016), pp. 239-250

[10] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 237, Springer, 2005 | Zbl

[11] Brady, Thomas; Watt, Colum K(π,1)’s for Artin groups of finite type, Geom. Dedicata, Volume 94 (2002) no. 1, pp. 225-250 Proceedings of the conference on geometric and combinatorial group theory, Part I. (Haifa) | DOI | MR | Zbl

[12] Brady, Thomas; Watt, Colum Non-crossing partition lattices in finite real reflection groups, Trans. Am. Math. Soc., Volume 360 (2008) no. 4, pp. 1983-2005 | DOI | MR | Zbl

[13] Ceballos, Cesar; Labbé, Jean-Philippe; Stump, Christian Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebr. Comb., Volume 39 (2014) no. 1, pp. 17-51 | DOI | MR | Zbl

[14] Chapoton, Frédéric Enumerative properties of generalized associahedra, Sémin. Lothar. Comb., Volume 51 (2004), B51b, 16 pages | MR | Zbl

[15] Chapoton, Frédéric Sur le nombre de réflexions pleines dans les groupes de Coxeter finis, Bull. Belg. Math. Soc. Simon Stevin, Volume 13 (2006) no. 4, pp. 585-596 | Zbl

[16] Cuntz, Michael; Stump, Christian On root posets for noncrystallographic root systems, Math. Comput., Volume 84 (2015) no. 291, pp. 485-503 | DOI | MR | Zbl

[17] Dyer, Matthew On the “Bruhat graph” of a Coxeter system, Compos. Math., Volume 78 (1991) no. 2, pp. 185-191 | MR | Zbl

[18] Fomin, Sergey; Reading, Nathan Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not., Volume 2005 (2005) no. 44, pp. 2709-2757 | DOI | MR | Zbl

[19] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | DOI | MR | Zbl

[20] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, 1990 | MR | Zbl

[21] Josuat-Vergès, Matthieu Refined enumeration of noncrossing chains and hook formulas, Ann. Comb., Volume 19 (2015) no. 3, pp. 443-460 | DOI | MR | Zbl

[22] Kreweras, Germain Sur les partitions non croisées d’un cycle, Discrete Math., Volume 1 (1972) no. 4, pp. 333-350 | DOI | Zbl

[23] Marsh, Robert; Reineke, Markus; Zelevinsky, Andrei Generalized associahedra via quiver representations, Trans. Am. Math. Soc., Volume 355 (2003) no. 10, pp. 4171-4186 | DOI | MR | Zbl

[24] Nica, Alexandru Non-crossing linked partitions, the partial order ≪ on NC n , and the S-transform, Proc. Am. Math. Soc., Volume 138 (2010) no. 4, pp. 1273-1285 | DOI | MR | Zbl

[25] Panyushev, Dmitri Ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra, Volume 274 (2004), pp. 822-846 | DOI | MR | Zbl

[26] Petrullo, Pasquale; Senato, Domenico Explicit formulae for Kerov polynomials, J. Algebr. Comb., Volume 33 (2011), pp. 141-151 | DOI | MR | Zbl

[27] Reading, Nathan Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Am. Math. Soc., Volume 359 (2007) no. 12, pp. 5931-5958 | DOI | MR | Zbl

[28] Reading, Nathan Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., Volume 29 (2015) no. 2, pp. 736-750 | DOI | MR | Zbl

[29] Reading, Nathan; Speyer, David Sortable elements in infinite Coxeter groups., Trans. Am. Math. Soc., Volume 363 (2011) no. 2, pp. 699-761 | DOI | MR | Zbl

[30] Speicher, Roland; Woroudi, Reza Boolean convolution, Free probability theory (Fields Institute Communications), Volume 12, American Mathematical Society, 1997, pp. 267-279 | MR | Zbl

[31] Thiel, Marko On the H-triangle of generalised nonnesting partitions, Eur. J. Comb., Volume 39 (2014), pp. 244-255 | DOI | MR | Zbl

Cited by Sources: