Abstract
Let Φ be an irreducible crystallographic rootsystem in a Euclidean space V, with Φ+ theset of positive roots. For α∈Φ, \(k \in Z\), let \(H(\alpha ,k)\) be the hyperplane \(\{ v \in V:\left\langle {\alpha ,v} \right\rangle = k\} \). We define a set of hyperplanes \(\mathcal{H} = \{ H(\delta ,1):\delta \in \Phi ^ + \} \cup \{ H(\delta ,0):\delta \in \Phi ^ + \} \). This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of \(\mathcal{H}\) is\(\left( {1 + ht} \right)^n \), where n is the rank of Φ and h is the Coxeter number of the finiteCoxeter group corresponding to Φ.
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Headley, P. On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups. Journal of Algebraic Combinatorics 6, 331–338 (1997). https://doi.org/10.1023/A:1008621126402
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DOI: https://doi.org/10.1023/A:1008621126402