On a family of hyperplane arrangements related to the affine Weyl groups
type: Item
instance of: scholarly article
title: On a family of hyperplane arrangements related to the affine Weyl groups
zbMATH Open document ID: 0911.52009
DOI: 10.1023/A:1008621126402
author: Patrick Headley
published in: Journal of Algebraic Combinatorics
publication date: 1999-04-28T00:00:00Z
review text: Let \(\Phi\) be an irreducible crystallographic root system in \({\mathbb R}^n\). The hyperplanes defined by \(\alpha(x)=0\) and \(\alpha(x)=1\), where \(\alpha\) ranges over the positive roots of \(\Phi\), form an arrangement \({\mathcal A}={\mathcal A}(\Phi)\). These arrangements were studied by \textit{J.-Y. Shi} [`The Kazhdan-Lusztig cells in certain affine Weyl groups' (1986; Zbl 0582.20030)], who showed that the number of connected components of \({\mathbb R}^n - \bigcup_{H \in {\mathcal A}} H\) (or ``regions'' of \(\mathcal A\)) is \((1+h)^n\), where \(h\) is the Coxeter number of \(\Phi\). In the article under review, this result is generalized. Let \(L\) be the partially ordered set of nonempty intersections of subsets of \(\mathcal A\), ordered by reverse inclusion, with minimal element \(0_L={\mathbb R}^n\). Let \(r(X)=n-\dim(X)\) and let \(\mu: L \times L \longrightarrow {\mathbb Z}\) be the Möbius function of \(L\). The Poincaré polynomial of \(\mathcal A\), defined by \[ P({\mathcal A},t):= \sum_{X \in L} \mu(0_L,X)(-t)^{r(X)}, \] has the property that \(P({\mathcal A},1)\) is the number of regions of \(\mathcal A\). For \({\mathcal A}={\mathcal A}(\Phi)\), the author shows that \[ P({\mathcal A},t)=(1+ht)^n. \] The proof is by induction on the rank of \(\Phi\), and uses Shi's result on the number of regions. The verification proceeds case-by-case, utilizing combinatorial identities for the classical root systems, and structural data compiled by \textit{P. Orlik} and \textit{L. Solomon} [Proc. Symp. Pure Math. 40, Part 2, 269-291 (1983; Zbl 0516.05019)] for the exceptional root systems. \textit{C. Athanasiadis} [Adv. Math. 122, No. 2, 193-233 (1996; Zbl 0872.52006)] has proven the same result, for the classical root systems, using a direct counting argument which is independent of Shi's result.
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 1078612
zbMATH Keywords: root system
zbMATH Keywords: affine Weyl group
zbMATH Keywords: Shi arrangement
zbMATH Keywords: characteristic polynomial
reviewed by: Michael Falk
MaRDI profile type: Publication
cites work: Characteristic polynomials of subspace arrangements and finite fields
cites work: Q4769056
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cites work: The Kazhdan-Lusztig cells in certain affine Weyl groups
cites work: Sign Types Corresponding to an Affine Weyl Group
cites work: Facing up to arrangements: face-count formulas for partitions of space by hyperplanes
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description: scientific article; zbMATH DE number 1078612