A symmetry of the descent algebra of a finite Coxeter group

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