A decomposition of the descent algebra of the hyperoctahedral group, II

Abstract

Elements of the hyperoctahedral group B n can be represented by lists of integers π = π 1 π 2 … π n , where the absolute values of the π's give a permutation of 1, …, n. The descent set of such a π is the set D( π) = { ifϵ [0 … n − 1]: π i > π i + 1 when i > 0, or π 1 < 0 when i = 0}. A descent class, in the group B n , is the collection of permutations with a given descent set. In [2] we have shown combinatorially a result of Solomon [12] stating that the product, in the group algebra Q [ B n ], of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q [ B n ]. We refer to this algebra here as Solomon's hyperoctahedral descent algebra and denote it by ∑ B n . The main goal of this paper is a decomposition of the multiplicative structure of ∑ B n . In particular, we obtain a complete set of minimal idempotents E λ (indexed by partitions of all k ⩽ n) and a basis of nilpotents for all the semi-ideals E μ ∑ B n E λ . To achieve this goal, it develops that ∑ B n acts on the so called B n-Lie monomials that were introduced in [5], developed in [2] but not fully understood until this paper. This action has a combinatorial description and is crucial in the construction of the idempotents and the nilpotents.