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On the Poincaré Series Associated with Coxeter Group Actions on Complements of Hyperplanes
Author(s) -
Lehrer G. I.
Publication year - 1987
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-36.2.275
Subject(s) - coxeter group , hyperplane , citation , combinatorics , coxeter complex , group (periodic table) , mathematics , series (stratigraphy) , library science , artin group , algebra over a field , computer science , pure mathematics , physics , paleontology , quantum mechanics , biology
Let W be a finite Coxeter group, realized as a group generated by reflections in the /-dimensional Euclidean space V. Let s/ be the hyperplane arrangement in C* = F(g)RC consisting of the complexifications of the reflecting hyperplanes of W in V. The hyperplane complement M = Mw = C l — {JHejl/H has been studied by Arnold [1] in the case when W = Sh Brieskorn [3], Deligne [7] and Orlik and Solomon [11, 12]. Denote by H*(M) the complex cohomology ring H*(M,C) of M. There is clearly an action of W on M, which transfers functorially to a W-action on H*(M). We shall be concerned with that action in this work. In particular we shall be concerned with the determination of the Poincare polynomials Pg(t), defined as follows. (1.1) DEFINITION. Let geW. The Poincare polynomial Pg(t) is defined by