A symmetry of the descent algebra of a finite Coxeter group

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 0 mapping D-W into the algebra of class functions of W. For W of type A. Jollenbeck 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. © 2004 Elsevier Inc. All rights reserved.