Résume | The Orlik-Solomon algebra of a finite Coxeter group W is the cohomology ring of the complement of the complexified reflection arrangement of W. As W-module it has the same dimension as the regular representation of W. In this talk I will discuss conjectural interrelated decompositions of these modules into induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of W. Some aspects of these decompositions have a long history. The decomposition of the group algebra of the symmetric group goes back to work on the free Lie algebra in the 1940s, the decomposition of the Orlik-Solomon algebra of the symmetric group was described by Lehrer and Solomon in 1986. Our approach provides a new common framework for both decompositions and an explicit connection between them, based on the representation theory of Solomon's descent algebra.
Our conjectures have been proved for dihedral and symmetric groups, and computationally verified for Coxeter groups of small rank. This is joint work with M. Bishop, J.M. Douglass and G. Roehrle. |