Orthogonal Idempotents in the Descent Algebra of Bn and Applications

We begin by briefly recalling some of our previous results on the descent algebra of the hyperoctahedral groups Bn. From this we construct a “nice” expression for the generating function of a family of orthogonal idempotents ρkn. More precisely ∑n k=1 ρknx = 1 2nn! ∑ π∈Bn (x−2d(π))↑ Bn π, where d(π) stands for the number of descents of π and (x)↑ Bn =(x+1)(x+3)(x+5)···(x+2n−1). We show that the dimension of the right ideal Q[Bn]ρ k n is given by the number c(n,k) of elements of Bn having k positive cycles. Thus the dimension is given by an analogue of the Stirling numbers of the first kind. We also show that the algebra Λ generated by the classes of elements of Bn having equal numbers of descents is commutative. This relates to Loday’s work on cyclic homology and Hochschild homology. In fact we show that we can define some λ operators on Λ which commute with the Hochschild boundary operator. This gives a decomposition for commutative hyperoctahedral algebra homology. We conclude our paper by presenting results about the hyperoctahedral shuffle algebra which are extensions of aspects of the work of Aldous, Bayer and Diaconis related to the shuffle algebra.