A symmetry of the descent algebra of a finite Coxeter group

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.