A symmetry of the descent algebra of a finite Coxeter group

Journal article

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.

Authors: Blessenohl, D; Hohlweg, C; Schocker, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 193
• Issue: 2
• Pages: 416–437
• Publication date: 2005-06-01
• Edition: Publisher's version
• DOI: 10.1016/j.aim.2004.05.007
• ISSN: 0001-8708
• Language: English
• Keywords: Parabolic subgroup; Minimal coset representative; Descent algebra; Coxeter group
• Subjects: Mathematics; Group theory and generalizations (mathematics)