Combinatorics and Poincaré polynomials of hyperplane complements for exceptional Weyl groups
type: Item
instance of: scholarly article
title: Combinatorics and Poincaré polynomials of hyperplane complements for exceptional Weyl groups
zbMATH Open document ID: 0838.20045
DOI: 10.1016/0097-3165(93)90060-L
author: Peter Fleischmann
author: Ingo Janiszczak
published in: Journal of Combinatorial Theory. Series A
publication date: 1994-12-06T00:00:00Z
review text: 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 \({\mat…
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 217948
zbMATH Keywords: indecomposable root systems
zbMATH Keywords: finite Coxeter groups
zbMATH Keywords: hyperplane complements
zbMATH Keywords: cohomology algebras
zbMATH Keywords: representations
zbMATH Keywords: exceptional Weyl groups
zbMATH Keywords: Poincaré polynomials
zbMATH Keywords: characteristic polynomials
zbMATH Keywords: intersections of hyperplanes
zbMATH Keywords: computer algebra system CAYLEY
zbMATH Keywords: Möbius function identities
reviewed by: Michael Falk
describes a project that uses: Cayley
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 217948