A symmetry of the descent algebra of a finite Coxeter group

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• 作者：Blessenohl ; Dieter ; Hohlweg ; Christophe ; Schocker ; Manfred
• 关键词：Coxeter group ; Descent algebra ; Parabolic subgroup ; Minimal coset representative
• 刊名：Advances in Mathematics
• 出版年：2005
• 出版时间：June 1, 2005
• 年：2005
• 卷：193
• 期：2
• 页码：416-437
• 全文大小：304 K

文摘

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.