Clusters, Coxeter-sortable elements and noncrossing partitions
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- by Nathan Reading PDF
- Trans. Amer. Math. Soc. 359 (2007), 5931-5958
Abstract:
We introduce Coxeter-sortable elements of a Coxeter group $W.$ For finite $W,$ we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in terms of their inversion sets and, in the classical cases, in terms of permutations.References
- Christos A. Athanasiadis, Thomas Brady, and Colum Watt, Shellability of noncrossing partition lattices, Proc. Amer. Math. Soc. 135 (2007), no. 4, 939–949. MR 2262893, DOI 10.1090/S0002-9939-06-08534-0
- I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33 (Russian). MR 0393065
- David Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 647–683 (English, with English and French summaries). MR 2032983, DOI 10.1016/j.ansens.2003.01.001
- Sara C. Billey and Tom Braden, Lower bounds for Kazhdan-Lusztig polynomials from patterns, Transform. Groups 8 (2003), no. 4, 321–332. MR 2015254, DOI 10.1007/s00031-003-0629-x
- Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3945–3975. MR 1401765, DOI 10.1090/S0002-9947-97-01838-2
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
- Thomas Brady and Colum Watt, A partial order on the orthogonal group, Comm. Algebra 30 (2002), no. 8, 3749–3754. MR 1922309, DOI 10.1081/AGB-120005817
- T. Brady and C. Watt, Non-crossing partition lattices in finite real reflection groups, to appear in Tran. Amer. Math. Soc.
- Frédéric Chapoton, Sergey Fomin, and Andrei Zelevinsky, Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), no. 4, 537–566. Dedicated to Robert V. Moody. MR 1941227, DOI 10.4153/CMB-2002-054-1
- M. J. Dyer, Hecke algebras and shellings of Bruhat intervals, Compositio Math. 89 (1993), no. 1, 91–115. MR 1248893
- Sergey Fomin and Andrei Zelevinsky, $Y$-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977–1018. MR 2031858, DOI 10.4007/annals.2003.158.977
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, DOI 10.1007/s00222-003-0302-y
- S. Fomin and N. Reading, Root systems and generalized associahedra, IAS/Park City Math. Ser., to appear.
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 0378456
- Robert Marsh, Markus Reineke, and Andrei Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4171–4186. MR 1990581, DOI 10.1090/S0002-9947-03-03320-8
- Jon McCammond, Noncrossing partitions in surprising locations, Amer. Math. Monthly 113 (2006), no. 7, 598–610. MR 2252931, DOI 10.2307/27642003
- Matthieu Picantin, Explicit presentations for the dual braid monoids, C. R. Math. Acad. Sci. Paris 334 (2002), no. 10, 843–848 (English, with English and French summaries). MR 1909925, DOI 10.1016/S1631-073X(02)02370-1
- Nathan Reading, Lattice congruences of the weak order, Order 21 (2004), no. 4, 315–344 (2005). MR 2209128, DOI 10.1007/s11083-005-4803-8
- Nathan Reading, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A 110 (2005), no. 2, 237–273. MR 2142177, DOI 10.1016/j.jcta.2004.11.001
- N. Reading, Cambrian Lattices, Adv. Math. 205 (2006), 313–353.
- N. Reading, Sortable elements and Cambrian lattices, to appear in Algebra Universalis.
- N. Reading and D. Speyer, Cambrian Fans, preprint, 2006 arXiv:math.CO/0606201.
- Victor Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), no. 1-3, 195–222. MR 1483446, DOI 10.1016/S0012-365X(96)00365-2
- Jian-Yi Shi, The enumeration of Coxeter elements, J. Algebraic Combin. 6 (1997), no. 2, 161–171. MR 1436533, DOI 10.1023/A:1008695121783
- C. Ingalls and H. Thomas, Generalized Catalan phenomena via representation theory of quivers, preprint, 2006 arXiv:math.RT/0612219.
Additional Information
- Nathan Reading
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
- Address at time of publication: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
- MR Author ID: 643756
- Email: nreading@umich.edu, nathan_reading@ncsu.edu
- Received by editor(s): August 18, 2005
- Published electronically: June 27, 2007
- Additional Notes: The author was partially supported by NSF grant DMS-0202430.
- © Copyright 2007 Nathan Reading
- Journal: Trans. Amer. Math. Soc. 359 (2007), 5931-5958
- MSC (2000): Primary 20F55; Secondary 05E15, 05A15
- DOI: https://doi.org/10.1090/S0002-9947-07-04319-X
- MathSciNet review: 2336311