A decomposition of the descent algebra of the hyperoctahedral group

Abstract

Elements of the hyperoctahedral group B n can be represented as lists of integers π = π 1 π 2 … π n , where the list of absolute value of the π k '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} ¦ π k > π k + 1} , where k = 0, we set π 0 = 0. A descent class A S of B n , is the formal sum of all elements of B n 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 [ B n ], 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 ∑ B n of Q [ B n ], as Solomon's Hyperoctahedral descent algebra. We derive from our combinatorial multiplication rule (for ∑ B n ) that there is a family of ideals J v ( n) of ∑ B n such that J 0 (n) ⊆- J 1 (n) … ⊆- J n (n) , with ∑ B n − v + 1 ∼- ∑ B n J v (n) . In [4], Garsia and Reutenauer relate a similar decomposition of the multiplicative structure of the descent algebra of the symmetric group S n , to the action of the symmetric group on the free Lie algebra. It develops that one can introduce a B n -analog of the free Lie algebra, and give similar results for ∑ B n .