Generalized descent algebras

Abstract

If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-$\mathbb{Z}$-module ${\mathcal{D}}_A(W)$ of the group algebra $\mathbb{Z}W$. We provide examples where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra and, if $W$ if of type $B$, the Mantaci-Reutenauer algebra.