A Semigroup Approach to Wreath-Product Extensions of Solomon's Descent Algebras

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 ΣGn associated with G ≀ Sn, the wreath product of the symmetric group Sn with an arbitrary group G. Techniques of Bidigare and Brown are adapted to construct an anti-homomorphism from the Sn-invariant subalgebra of the semigroup algebra of Σ G n into the group algebra of G ≀ Sn. The generalized descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when G is abelian.