A decomposition of the descent algebra of the hyperoctahedral group

AbstractElements of the hyperoctahedral group Bn can be represented as lists of integers π = π1π2 … πn, where the list of absolute value of the πk's is a permutation of the integers from 1 to n. The descent set of such a π is Des(π) = {kϵ {0, …, n − 1} ¦ πk > πk + 1}, where k = 0, we set π0 = 0. A descent class AS of Bn, is the formal sum of all elements of Bn with a given descent set S ⊆{0, 1, …, n − 1}. In this paper, we shall give a combinatorial proof of a result of Solomon [6] stating that the product, in the group algebra Q[Bn], of two descent classes is a linear combination of descent classes. We refer to the linear span of these descent classes (it is a subalgebra ∑Bn of Q[Bn], as Solomon's Hyperoctahedral descent algebra. We derive from our combinatorial multiplication rule (for ∑ Bn) that there is a family of ideals Jv(n) of ∑ Bn such that J0(n) ⊆- J1(n) … ⊆- Jn(n), with ∑ Bn − v + 1 ∼- ∑ BnJv(n). In [4], Garsia and Reutenauer relate a similar decomposition of the multiplicative structure of the descent algebra of the symmetric group Sn, to the action of the symmetric group on the free Lie algebra. It develops that one can introduce a Bn-analog of the free Lie algebra, and give similar results for ∑ Bn