A DECOMPOSITION OF THE DESCENT ALGEBRA OF THE HYPEROCTAHEDRAL GROUP .2.

Elements of the hyperoctahedral group Bn can be represented by lists of integers π = π1 π2 ... πn, 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{lunate} [0 ... n - 1]: πi > πi + 1 when i > 0, or π1 < 0 when i = 0}. A descent class, in the group Bn, 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[Bn], of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q[Bn]. We refer to this algebra here as Solomon's hyperoctahedral descent algebra and denote it by ∑ Bn. The main goal of this paper is a decomposition of the multiplicative structure of ∑ Bn. In particular, we obtain a complete set of minimal idempotents Eλ (indexed by partitions of all k ≤ n) and a basis of nilpotents for all the semi-ideals Eμ ∑ BnEλ. To achieve this goal, it develops that ∑ Bn acts on the so called Bn-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. © 1992.