Noncrossing partitions, Bruhat order and the cluster complex
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:
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Accepted:
Published online:
DOI: 10.5802/aif.3294
Classification: 05A15,  20F55
Keywords: Finite Coxeter groups, Noncrossing partitions, Bruhat order, cluster complex
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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, Tome 202, American Mathematical Society, 2009 no. 949, x+159 pages | MR: 2561274 | Zbl: 1191.05095

[2] Athanasiadis, Christos A. On some enumerative aspects of generalized associahedra, Eur. J. Comb., Tome 28 (2007), pp. 1208-1215 | Article | MR: 2305586 | Zbl: 1117.52013

[3] Athanasiadis, Christos A.; Brady, Thomas; McCammond, Jon; Watt, Colum h-vectors of generalized associahedra and noncrossing partitions, Int. Math. Res. Not., Tome 2006 (2006) no. 12, 69705, 28 pages | MR: 2249994 | Zbl: 1112.20032

[4] Athanasiadis, Christos A.; Savvidou, Christina The local h-vector of the cluster subdivision of a simplex, Sémin. Lothar. Comb., Tome 66 (2012), B66c, 21 pages | MR: 2971012 | Zbl: 1253.05150

[5] Barnard, Emily; Reading, Nathan Coxeter-biCatalan combinatorics, J. Algebr. Comb., Tome 47 (2018) no. 2, pp. 241-300 | Article | MR: 3775222 | Zbl: 1387.05270

[6] Belinschi, Serban T.; Nica, Alexandru η-series and and a boolean Bercovici-Pata bijection for bounded k-tuples, Adv. Math., Tome 217 (2008), pp. 1-41 | Article | MR: 2357321 | Zbl: 1129.46052

[7] Bessis, David The dual braid monoid, Ann. Sci. Éc. Norm. Supér., Tome 36 (2003) no. 5, pp. 647-683 | Article | MR: 2032983 | Zbl: 1064.20039

[8] Biane, Philippe Some properties of crossings and partitions, Discrete Math., Tome 175 (1997) no. 1-3, pp. 41-53 | Article | MR: 1475837 | Zbl: 0892.05006

[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, Tome 237, Springer, 2005 | Zbl: 1110.05001

[11] Brady, Thomas; Watt, Colum K(π,1)’s for Artin groups of finite type, Geom. Dedicata, Tome 94 (2002) no. 1, pp. 225-250 (Proceedings of the conference on geometric and combinatorial group theory, Part I. (Haifa)) | Article | MR: 1950880 | Zbl: 1053.20034

[12] Brady, Thomas; Watt, Colum Non-crossing partition lattices in finite real reflection groups, Trans. Am. Math. Soc., Tome 360 (2008) no. 4, pp. 1983-2005 | Article | MR: 2366971 | Zbl: 1187.20051

[13] Ceballos, Cesar; Labbé, Jean-Philippe; Stump, Christian Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebr. Comb., Tome 39 (2014) no. 1, pp. 17-51 | Article | MR: 3144391 | Zbl: 1286.05180

[14] Chapoton, Frédéric Enumerative properties of generalized associahedra, Sémin. Lothar. Comb., Tome 51 (2004), B51b, 16 pages | MR: 2080386 | Zbl: 1160.05342

[15] Chapoton, Frédéric Sur le nombre de réflexions pleines dans les groupes de Coxeter finis, Bull. Belg. Math. Soc. Simon Stevin, Tome 13 (2006) no. 4, pp. 585-596 | Zbl: 1150.20023

[16] Cuntz, Michael; Stump, Christian On root posets for noncrystallographic root systems, Math. Comput., Tome 84 (2015) no. 291, pp. 485-503 | Article | MR: 3266972 | Zbl: 1337.06001

[17] Dyer, Matthew On the “Bruhat graph” of a Coxeter system, Compos. Math., Tome 78 (1991) no. 2, pp. 185-191 | MR: 1104786 | Zbl: 0784.20019

[18] Fomin, Sergey; Reading, Nathan Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not., Tome 2005 (2005) no. 44, pp. 2709-2757 | Article | MR: 2181310 | Zbl: 1117.52017

[19] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras II. Finite type classification, Invent. Math., Tome 154 (2003) no. 1, pp. 63-121 | Article | MR: 2004457 | Zbl: 1054.17024

[20] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Tome 29, Cambridge University Press, 1990 | MR: 1066460 | Zbl: 0725.20028

[21] Josuat-Vergès, Matthieu Refined enumeration of noncrossing chains and hook formulas, Ann. Comb., Tome 19 (2015) no. 3, pp. 443-460 | Article | MR: 3395490 | Zbl: 1327.05027

[22] Kreweras, Germain Sur les partitions non croisées d’un cycle, Discrete Math., Tome 1 (1972) no. 4, pp. 333-350 | Article | Zbl: 0231.05014

[23] Marsh, Robert; Reineke, Markus; Zelevinsky, Andrei Generalized associahedra via quiver representations, Trans. Am. Math. Soc., Tome 355 (2003) no. 10, pp. 4171-4186 | Article | MR: 1990581 | Zbl: 1042.52007

[24] Nica, Alexandru Non-crossing linked partitions, the partial order on NC n , and the S-transform, Proc. Am. Math. Soc., Tome 138 (2010) no. 4, pp. 1273-1285 | Article | MR: 2578521 | Zbl: 1201.46060

[25] Panyushev, Dmitri Ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra, Tome 274 (2004), pp. 822-846 | Article | MR: 2043377 | Zbl: 1067.17005

[26] Petrullo, Pasquale; Senato, Domenico Explicit formulae for Kerov polynomials, J. Algebr. Comb., Tome 33 (2011), pp. 141-151 | Article | MR: 2747805 | Zbl: 1245.05134

[27] Reading, Nathan Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Am. Math. Soc., Tome 359 (2007) no. 12, pp. 5931-5958 | Article | MR: 2336311 | Zbl: 1189.05022

[28] Reading, Nathan Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., Tome 29 (2015) no. 2, pp. 736-750 | Article | MR: 3335492 | Zbl: 1314.05015

[29] Reading, Nathan; Speyer, David Sortable elements in infinite Coxeter groups., Trans. Am. Math. Soc., Tome 363 (2011) no. 2, pp. 699-761 | Article | MR: 2728584 | Zbl: 1231.20036

[30] Speicher, Roland; Woroudi, Reza Boolean convolution, Free probability theory (Fields Institute Communications) Tome 12, American Mathematical Society, 1997, pp. 267-279 | MR: 1426845 | Zbl: 0872.46033

[31] Thiel, Marko On the H-triangle of generalised nonnesting partitions, Eur. J. Comb., Tome 39 (2014), pp. 244-255 | Article | MR: 3168529 | Zbl: 1284.05336

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