A divisibility result in combinatorics of generalized braids

For every finite Coxeter group Γ, each positive braid in the corresponding braid group admits a unique decomposition as a finite sequence of elements of Γ, the so-called Garside-normal form. The study of the associated adjacency matrix Adj(Γ) allows to count the number of Garside-normal form of a given length. In this paper we prove that the characteristic polynomial of Adj(Bn) divides the one of Adj(Bn+1). The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type A. We observe that this does not hold for type D. The other Coxeter types (I, E, F and H) are also studied.