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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On a refinement of the generalized Catalan numbers for Weyl groups


Author: Christos A. Athanasiadis
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 179-196
MSC (2000): Primary 20F55; Secondary 05E99, 20H15
Published electronically: March 23, 2004
MathSciNet review: 2098091
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Abstract: Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$, spanning a Euclidean space $V$, and let $m$ be a positive integer. It is known that the set of regions into which the fundamental chamber of $W$ is dissected by the hyperplanes in $V$ of the form $(\alpha, x) = k$ for $\alpha \in \Phi$ and $k = 1, 2,\dots,m$ is equinumerous to the set of orbits of the action of $W$ on the quotient $\check{Q} / \, (mh+1) \, \check{Q}$. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of $\Phi$, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case $m=1$ and $\Phi = A_{n-1}$.


References [Enhancements On Off] (What's this?)

  • 1. C.A. Athanasiadis, Deformations of Coxeter hyperplane arrangements and their characteristic polynomials, in ``Arrangements - Tokyo 1998'' (M. Falk and H. Terao, eds.), Adv. Stud. Pure Math. 27, Kinokuniya, Tokyo, 2000, pp. 1-26. MR 2001i:52035
  • 2. C.A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes, Bull. London Math. Soc. (to appear).
  • 3. D. Bessis, The dual braid monoid, Ann. Sci. Ecole Norm. Sup. 36 (2003), 647-683.
  • 4. P. Cellini and P. Papi, $ad$-nilpotent ideals of a Borel subalgebra, J. Algebra 225 (2000), 130-141. MR 2001g:17017
  • 5. P. Cellini and P. Papi, $ad$-nilpotent ideals of a Borel subalgebra II, J. Algebra 258 (2002), 112-121.
  • 6. F. Chapoton, S. Fomin and A.V. Zelevinsky, Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), 537-566. MR 2003j:52014
  • 7. D.Z. Djokovic, On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups, Proc. Amer. Math. Soc. 80 (1980), 181-184. MR 81b:20052
  • 8. S. Fomin and A.V. Zelevinsky, $Y$-systems and generalized associahedra, Ann. of Math. 158 (2003), 977-1018.
  • 9. S. Fomin and A.V. Zelevinsky, Cluster algebras II: finite type classification, Invent. Math. 154 (2003), 63-121.
  • 10. M.D. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), 17-76. MR 95a:20014
  • 11. J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, England, 1990. MR 92h:20002
  • 12. T. Venkata Narayana, Sur les treillis formés par les partitions d'un entier et leurs applications à la théorie des probabilités, Comp. Rend. Acad. Sci. Paris 240 (1955), 1188-1189. MR 17:14a
  • 13. T. Venkata Narayana, Lattice path combinatorics with statistical applications, University of Toronto Press, 1979. MR 81f:60019
  • 14. D.I. Panyushev, Ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra (to appear).
  • 15. M. Picantin, Explicit presentations for the dual braid monoids, C. R. Math. Acad. Sci. Paris Sér. I Math. 334 (2002), 843-848. MR 2003d:20046
  • 16. A. Postnikov and R.P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A 91 (2000), 544-597. MR 2003g:52032
  • 17. V. Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), 195-222. MR 99f:06005
  • 18. V. Reiner and V. Welker, On the Charney-Davis and Neggers-Stanley conjectures, preprint, 2002, 38 pages.
  • 19. J.-Y. Shi, Alcoves corresponding to an affine Weyl group, J. London Math. Soc. 35 (1987), 42-55. MR 88g:20103a
  • 20. J.-Y. Shi, Sign types corresponding to an affine Weyl group, J. London Math. Soc. 35 (1987), 56-74. MR 88g:20103b
  • 21. J.-Y. Shi, The number of $\oplus$-sign types, Quart. J. Math. Oxford Ser. (2) 48 (1997), 93-105. MR 98c:20080
  • 22. E. Sommers, $B$-stable ideals in the nilradical of a Borel subalgebra, ArXiV preprint math.RT/0303182, March 2003.
  • 23. R.P. Stanley, Hyperplane arrangements, parking functions and tree inversions, in ``Mathematical Essays in Honor of Gian-Carlo Rota'' (B.E. Sagan and R.P. Stanley, eds.), Progress in Math. 161 (1998), Birkhäuser, Boston, pp. 359-375. MR 99f:05006
  • 24. R.A. Sulanke, Catalan path statistics having the Narayana distribution, Discrete Math. 180 (1998), 369-389. MR 99b:05004

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Additional Information

Christos A. Athanasiadis
Affiliation: Department of Mathematics, University of Crete, 71409 Heraklion, Crete, Greece
Email: caa@math.uoc.gr

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03548-2
PII: S 0002-9947(04)03548-2
Keywords: Weyl group, coroot lattice, root order, ideals, hyperplane arrangements, regions, Narayana numbers
Received by editor(s): March 16, 2003
Received by editor(s) in revised form: June 26, 2003
Published electronically: March 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society