The Flag Descent Algebra and the Colored Eulerian Descent Algebra

摘要

We prove that the group algebra of the hyperoctahedral group contains asubalgebra corresponding to the flag descent number of Adin, Brenti, andRoichman. This algebra is in fact the span of the basis elements of the type Aand type B Eulerian descent algebras. We describe a set of orthogonalidempotents which spans the flag descent algebra and prove that it contains thetype A Eulerian descent algebra as a two-sided ideal. Using a new coloredanalogue of Stanley's $P$-partitions, we prove the existence of a coloredEulerian descent algebra which is a subalgebra of the Mantaci-Reutenaueralgebra. We also describe a set of orthogonal idempotents that spans thecolored Eulerian descent algebra and includes, as a special case, the familiarEulerian idempotents in the group algebra of the symmetric group.