On a family of hyperplane arrangements related to the affine Weyl groups
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.
scientific article; zbMATH DE number 1078612
scientific article