The colored Eulerian descent algebra

Abstract

Using a new colored analogue of $$P$$P-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci---Reutenauer algebra. This algebra has a basis consisting of formal sums of colored permutations with the same number of descents (using Steingrimsson's definition of the descent set of a colored permutation). The colored Eulerian descent algebra extends familiar Eulerian descent algebras from the symmetric group algebra and the hyperoctahedral group algebra to colored permutation group algebras. We also describe orthogonal idempotents that span the colored Eulerian descent algebra and include, as a special case, the familiar Eulerian idempotents in the group algebra of the symmetric group.