The Colored Eulerian Descent Algebra

TL;DR

This paper introduces a new colored Eulerian descent algebra within the Mantaci-Reutenauer algebra, extending classical Eulerian descent algebras to colored permutations using a novel P-partition approach.

Contribution

It constructs a colored Eulerian descent algebra as a subalgebra with a basis of colored permutations grouped by descents, generalizing known Eulerian algebras.

Findings

Established the existence of the colored Eulerian descent algebra.

Provided a basis of formal sums of colored permutations with the same number of descents.

Described a set of orthogonal idempotents including classical Eulerian idempotents.

Abstract

Using a new colored analogue of 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 Steingr\'imsson'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 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.