The descent algebra
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 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
, where class functions of
W have been extended to the group algebra of
W linearly. They conjectured that this symmetry property of
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 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.