## Shellability of noncrossing partition lattices

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- by Christos A. Athanasiadis, Thomas Brady and Colum Watt PDF
- Proc. Amer. Math. Soc.
**135**(2007), 939-949 Request permission

## Abstract:

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type $D_n$ and those of exceptional type and rank at least three.## References

- Christos A. Athanasiadis,
*On a refinement of the generalized Catalan numbers for Weyl groups*, Trans. Amer. Math. Soc.**357**(2005), no. 1, 179–196. MR**2098091**, DOI 10.1090/S0002-9947-04-03548-2 - Christos A. Athanasiadis and Victor Reiner,
*Noncrossing partitions for the group $D_n$*, SIAM J. Discrete Math.**18**(2004), no. 2, 397–417. MR**2112514**, DOI 10.1137/S0895480103432192 - 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 - Anders Björner,
*Shellable and Cohen-Macaulay partially ordered sets*, Trans. Amer. Math. Soc.**260**(1980), no. 1, 159–183. MR**570784**, DOI 10.1090/S0002-9947-1980-0570784-2 - Anders Björner and Francesco Brenti,
*Combinatorics of Coxeter groups*, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR**2133266** - Thomas Brady,
*A partial order on the symmetric group and new $K(\pi ,1)$’s for the braid groups*, Adv. Math.**161**(2001), no. 1, 20–40. MR**1857934**, DOI 10.1006/aima.2001.1986 - Thomas Brady and Colum Watt,
*$K(\pi ,1)$’s for Artin groups of finite type*, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 225–250. MR**1950880**, DOI 10.1023/A:1020902610809 - T. Brady and C. Watt,
*Lattices in finite real reflection groups*, preprint, 2005, 29 pages, Trans. Amer. Math. Soc. (to appear). - Frédéric Chapoton,
*Enumerative properties of generalized associahedra*, Sém. Lothar. Combin.**51**(2004/05), Art. B51b, 16. MR**2080386** - 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 - 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 - G. Kreweras,
*Sur les partitions non croisées d’un cycle*, Discrete Math.**1**(1972), no. 4, 333–350 (French). MR**309747**, DOI 10.1016/0012-365X(72)90041-6 - J. McCammond,
*Noncrossing partitions in surprising locations*, preprint, 2003, 14 pages, Amer. Math. Monthly (to appear). - J. McCammond,
*An introduction to Garside structures*, preprint, 2004, 28 pages. - Dmitri I. Panyushev,
*ad-nilpotent ideals of a Borel subalgebra: generators and duality*, J. Algebra**274**(2004), no. 2, 822–846. MR**2043377**, DOI 10.1016/j.jalgebra.2003.09.007 - 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 - V. Reiner, personal communication with the first author, 2002.
- Eric N. Sommers,
*$B$-stable ideals in the nilradical of a Borel subalgebra*, Canad. Math. Bull.**48**(2005), no. 3, 460–472. MR**2154088**, DOI 10.4153/CMB-2005-043-4 - Richard P. Stanley,
*Enumerative combinatorics. Vol. 1*, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR**1442260**, DOI 10.1017/CBO9780511805967 - Robert Steinberg,
*Finite reflection groups*, Trans. Amer. Math. Soc.**91**(1959), 493–504. MR**106428**, DOI 10.1090/S0002-9947-1959-0106428-2

## Additional Information

**Christos A. Athanasiadis**- Affiliation: Department of Mathematics, University of Crete, 71409 Heraklion, Crete, Greece
- Address at time of publication: Department of Mathematics, University of Athens, Panepistimioupolis, Athens 15784, Greece
- MR Author ID: 602846
- Email: caath@math.uoa.gr
**Thomas Brady**- Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
- Email: tom.brady@dcu.ie
**Colum Watt**- Affiliation: School of Mathematics, Dublin Institute of Technology, Dublin 8, Ireland
- Email: colum.watt@dit.ie
- Received by editor(s): August 1, 2005
- Received by editor(s) in revised form: October 25, 2005
- Published electronically: September 26, 2006
- Additional Notes: This work was supported in part by the American Institute of Mathematics (AIM) and the NSF
- Communicated by: John R. Stembridge
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**135**(2007), 939-949 - MSC (2000): Primary 20F55; Secondary 05E15, 05E99, 06A07
- DOI: https://doi.org/10.1090/S0002-9939-06-08534-0
- MathSciNet review: 2262893