Generalized Poincaré series for models¶of the braid arrangements

Abstract:

Let be the complexified Coxeter arrangement of hyperplanes of type A _{n −1} (n≥ 3). It is well known that the “minimal” projective De Concini–Procesi model of is isomorphic to the moduli space of stable n plus;1-pointed curves of genus 0. In this paper we study, from the point of view of models of arrangements, the action of the symmetric group Σ _n on the integer cohomology ring of . In fact we find a formula for the generalized Poincaré series which encodes all the information about this representation of Σ _n . This formula, which is obtained by using the elementary combinatorial properties of a ℤ-basis of and turns out to be very direct, should be compared with a more general result due to Getzler (see [5]).