Hostname: page-component-6bf8c574d5-vmclg Total loading time: 0 Render date: 2025-02-26T15:06:51.883Z Has data issue: false hasContentIssue false
  • English
  • Français

A vanishing theorem for hyperplane cohomology

Published online by Cambridge University Press:  17 April 2009

G.I. Lehrer
G.I. Lehrer
Affiliation:
School of Mathematics and StatisticsUniversity of SydneySydney NSW 2006Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A be a hyperplane arrangement in an arbitrary finite dimensional vector space V and let G ≤ GL(V) be an automorphism group of A. If λ is a complex representation of G such that (λ,1)GH=0 for all pointwise isotropy groups GH (H ∈ A), then we prove the “local-global” result that λ does not appear in the representation of G on the Orlik-Solomon algebra of A. The result is applied to complex reflection groups and to finite orthogonal groups. It may also be viewed as a combinatorial result concerning the homology of the lattice of intersections of A. A more general version of the main result is also discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 53 , Issue 3 , June 1996 , pp. 361 - 368
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Broué, M. and Malle, G., ‘Zyklotomische Heckealgebren’, Ast´erisque 212 (1993), 119–189.Google Scholar
[2]Hanlon, P., ‘A proof of a conjecture of Stanley concerning partitions of a set’, European J. Combin. 4 (1983), 137–141.Google Scholar
[3]Lehrer, G.I., ‘Rational tori, semisimple orbits and the topology of hyperplane complements’, Comment. Math. Helv. 67 (1992), 226–251.Google Scholar
[4]Lehrer, G.I., ‘The ℓ-adic cohomology of hyperplane complements’, Bull. Lond. Math. Soc. 24 (1992), 76–82.Google Scholar
[5]Lehrer, G.I. and Rylands, L., ‘The split building of a reductive group’, Math. Ann. 296 (1993), 607–624.Google Scholar
[6]Lehrer, G.I. and Solomon, L., ‘On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes’, J. Algebra 104 (1986), 410–424.CrossRefGoogle Scholar
[7]Orlik, P. and Solomon, L., ‘Combinatorics and the topology of complements of hyperplanes’, Invent. Math. 56 (1980), 167–189.Google Scholar
[8]Orlik, P. and Terao, H., Arrangements of hyperplanes (Springer-Verlag, Berlin, Heidelberg, New York, 1992).Google Scholar
[9]Rota, G.-C., ‘On the foundations of combinatorial theory I. Theory of Möbius functions’, Z. Wahrsch. 2 (1964), 340–368.Google Scholar
[10]Stanley, R., ‘Some aspects of groups acting on finite posets’, J. Comb. Theory Ser. A 32 (1982), 132–161.CrossRefGoogle Scholar
[11]Taylor, D.E., The geometry of the classical groups, Sigma Series in Pure Mathematics, 9 (Heldermann Verlag, Berlin, 1992).Google Scholar