×

Combinatorics and Poincaré polynomials of hyperplane complements for exceptional Weyl groups. (English) Zbl 0838.20045

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 G. I. Lehrer and L. Solomon [J. Algebra 104, 410-424 (1986; Zbl 0608.20010)], 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 \({\mathcal P}_{W,w} (t)\).

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20G05 Representation theory for linear algebraic groups
20C40 Computational methods (representations of groups) (MSC2010)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)

Software:

Cayley
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brieskorn, E., Sur les groupes de tresses (d’après V. I. Arnold), (Séminaire Bourbaki 24e année 1971/72. Séminaire Bourbaki 24e année 1971/72, Lecture Notes in Mathematics (1973), Springer: Springer Berlin), No. 317
[2] Carter, R., Conjugacy classes in the Weyl group, Compositio Math., 25, 1-59 (1972) · Zbl 0254.17005
[3] Carter, R., Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (1985), Wiley: Wiley New York · Zbl 0567.20023
[4] Fleischmann, P.; Janiszczak, I., The lattices and Moebius functions of closed subrootsystems and hyperplane complements for classical Weyl groups, Manuscripta Math., 72, 375-403 (1991) · Zbl 0790.52006
[5] P. Fleischmann and I. JaniszczakJ. Algebra; P. Fleischmann and I. JaniszczakJ. Algebra · Zbl 0809.20007
[6] Fleischmann, P.; Janiszczak, I., The semisimple conjugacy classes of finite groups of Lie type \(E_6\) and \(E_7\), Comm. in Algebra, 21, 1, 93-161 (1993) · Zbl 0813.20015
[7] Fleischmann, P.; Janiszczak, I., The semisimple conjugacy classes and the generic class number of the finite simple groups of Lie type \(E_8 (1992)\), Institut für Experimentelle Mathematik: Institut für Experimentelle Mathematik Essen, preprint No. 17
[8] Lehrer, G. I., On the Poincaré series associated with Coxeter group actions on complements of hyperplanes, J. London Math. Soc. (2), 36, No. 2, 275-294 (1987) · Zbl 0649.20041
[9] Lehrer, G. I., On Hyperoctahedral Hyperplane Complements, (Proceedings of Symposia in Pure Mathematics, Vol. 47 (1987), Amer. Math. Soc: Amer. Math. Soc Providence, Rhode Island), Part 2 · Zbl 0653.20045
[10] Lehrer, G. I.; Solomon, L., On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra, 104, 410-424 (1986) · Zbl 0608.20010
[11] Orlik, P.; Solomon, L., Combinatories and topology of complements of hyperplanes, Invent. Math., 56, 167-189 (1980) · Zbl 0432.14016
[12] Orlik, P.; Solomon, L., Unitary reflection groups and cohomology, Invent. Math., 59, 77-94 (1980) · Zbl 0452.20050
[13] Rota, G.-C, On the foundations of combinatorial theory. I. Möbius functions, Z. Wahrscheinlichkeitstheorie, 2, 340-368 (1964) · Zbl 0121.02406
[14] Steinberg, R., Torsion in reductive groups, Adv. Math., 15, 63-92 (1975) · Zbl 0312.20026
[15] Steinberg, R., Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc., 112, 392-400 (1964) · Zbl 0196.39202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.