Combinatorics and Poincaré polynomials of hyperplane complements for exceptional Weyl groups
Let \(\Phi\) be an indecomposable root system, with associated finite Coxeter group \(W\) acting on a complexified real vector space \(V\). For \(\alpha\in \Phi\) let \(H_\alpha\) denote the hyperplane in \(V\) fixed by the reflection \(s_\alpha\). Then \(W\) acts on the hyperplane complement \(M= V- \bigcup_{\alpha\in \Phi} H_\alpha\), inducing a representation of \(W\) on the cohomology algebra \(H^* (M, \mathbb{C})\). In the paper under review, these representations are computed, in a sense described below, for all the exceptional Weyl groups. The classical Weyl groups were treated previously by \textit{G. I. Lehrer} and \textit{L. Solomon} [J. Algebra 104, 410-424 (1986; Zbl 0608.20010)], \textit{G. I. Lehrer} [Proc. Symp. Pure Math. 47, Part 2, 219-234 (1987; Zbl 0653.20045)]\ and the present authors [Manuscr. Math. 72, 375-403 (1991; Zbl 0790.52006)]. For \(w\in W\) define the Poincaré polynomial \({\mathcal P}_{W,w} (t):= \sum_{i=0}^\infty \text{trace} (w, H^i (M, \mathbb{C})) \cdot t^i\). The authors compute and tabulate these polynomials for \(W\) of type \(G_2\), \(F_4\), \(E_6\), \(E_7\), and \(E_8\), and representatives \(w\) of each conjugacy class in \(W\). The starting point is a formula, due to Orlik and Solomon, relating \({\mathcal P}_{W,w} (t)\) to the characteristic polynomial of the poset consisting of those intersections of hyperplanes \(H_\alpha\) which are stabilized by \(w\). The computation of these characteristic polynomials for the exceptional Weyl groups is implemented on computer, using the computer algebra system CAYLEY along with some additional routines written by the authors. A result of Steinberg is used to reduce the ground field to \(\text{GF}(p)\), where \(p\) is a ``good'' prime, facilitating the computation. In the first section of the paper, the authors use Möbius function identities to derive some known results about these representations, and to give combinatorial explanations for certain of the integer roots of the \({\mathcal P}_{W,w} (t)\).
scientific article; zbMATH DE number 217948
scientific article