A semigroup approach to wreath-product extensions of Solomon's descent algebras

Abstract

There is a well-known combinatorial definition, based on ordered set partitions, of the semigroup of faces of the braid arrangement. We generalize this definition to obtain a semigroup Sigma_n^G associated with G wr S_n, the wreath product of the symmetric group S_n with an arbitrary group G. Techniques of Bidigare and Brown are adapted to construct an anti-homomorphism from the S_n-invariant subalgebra of the semigroup algebra of Sigma_n^G into the group algebra of G wr S_n. The generalized descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when G is abelian.