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Transactions of the American Mathematical Society

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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 $\Phi = A_{n-1}$.

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Additional Information

Christos A. Athanasiadis
Affiliation: Department of Mathematics, University of Crete, 71409 Heraklion, Crete, Greece

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