| We prove that the group algebra of the hyperoctahedral group contains a subalgebra corresponding to the flag descent number of Adin, Brenti, and Roichman. This algebra is in fact the span of the basis elements of the type A and type B Eulerian descent algebras. We describe a set of orthogonal idempotents which span the flag descent algebra and prove that it contains the type A Eulerian descent algebra as a two-sided ideal.;Using a new colored analogue of Stanley's P-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algebra. We also describe a set of orthogonal idempotents that spans the colored Eulerian descent algebra and includes, as a special case, the familiar Eulerian idempotents in the group algebra of the symmetric group. |
