A symmetry of the descent algebra of a finite Coxeter group

Abstract

The descent algebra D W of a finite Coxeter group W, discovered by Solomon in 1976, is a subalgebra of the group algebra of W. Due to Solomon, it is intimately linked to the representation theory of W, by means of a homomorphism of algebras θ mapping D W into the algebra of class functions of W. For W of type A, Jöllenbeck and Reutenauer derived the identity θ ( X ) ( Y ) = θ ( Y ) ( X ) for all X , Y ∈ D W , where class functions of W have been extended to the group algebra of W linearly. They conjectured that this symmetry property of D W holds for arbitrary finite Coxeter groups W. This conjecture—actually a combinatorial refinement—is proven here. As a consequence, several properties of the characters of W afforded by the primitive idempotents of D W may be derived at once, including a symmetry of the corresponding character table, and a combinatorial description of their intertwining numbers with the descent characters of W. This recovers and extends results of Gessel-Reutenauer and Scharf-Thibon on the symmetric group, and of Poirier on the hyperoctahedral group.