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GENERALIZED CATALAN NUMBERS, WEYL GROUPS AND ARRANGEMENTS OF HYPERPLANES

  • CHRISTOS A. ATHANASIADIS (a1)
Abstract

For an irreducible, crystallographic root system $\Phi$ in a Euclidean space $V$ and a positive integer $m$, the arrangement of hyperplanes in $V$ given by the affine equations $(\alpha, x)\,{=}\,k$, for $\alpha\,{\in}\,\Phi$ and $k\,{=}\,0, 1,\dots,m$, is denoted here by ${\mathcal A}_{\Phi}^m$. The characteristic polynomial of ${\mathcal A}_{\Phi}^m$ is related in the paper to that of the Coxeter arrangement ${\mathcal A}_{\Phi}$ (corresponding to $m\,{=}\,0$), and the number of regions into which the fundamental chamber of ${\mathcal A}_{\Phi}$ is dissected by the hyperplanes of ${\mathcal A}_{\Phi}^m$ is deduced to be equal to the product $\prod_{i=1}^{\ell} ({e_i\,{+}\,m h\,{+}\,1})/({e_i\,{+}\,1})$, where $e_1, e_2,\dots,e_\ell$ are the exponents of $\Phi$ and $h$ is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of $\Phi$ are included.

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Bulletin of the London Mathematical Society
  • ISSN: 0024-6093
  • EISSN: 1469-2120
  • URL: /core/journals/bulletin-of-the-london-mathematical-society
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