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

Abstract

There is a well-known combinatorial model, based on ordered set partitions, of the semigroup of faces of the braid arrangement. We generalize this model to obtain a semigroup ${\cal F}_n^G$ 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 ${\cal F}_n^G$ into the group algebra of $G\wr S_n$. The colored descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when $G$ is abelian.