 
Summary: UNITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS
MARCELO AGUIAR, JEANCHRISTOPHE NOVELLI, JEANYVES THIBON
Abstract. We construct unital extensions of the higher order peak algebras de
fined by Krob and the third author in [Ann. Comb. 9 (2005), 411430.], and
show that they can be obtained as homomorphic images of certain subalgebras of
the MantaciReutenauer algebras of type B. This generalizes a result of Bergeron,
Nyman and the first author [Trans. AMS 356 (2004), 27812824.].
1. Introduction
A descent of a permutation Sn is an index i such that (i) > (i + 1). A
descent is a peak if moreover i > 1 and (i) > (i  1). The sums of permutations
with a given descent set span a subalgebra of the group algebra, the descent algebra
n. The peak algebra °Pn of Sn is a subalgebra of its descent algebra, spanned by
sums of permutations having the same peak set. This algebra has no unit.
Descent algebras can be defined for all finite Coxeter groups [19]. In [2], it is shown
that the peak algebra of Sn can be naturally extended to a unital algebra, which
is obtained as a homomorphic image of the descent algebra of the hyperoctahedral
group Bn.
As explained in [5], it turns out that a fair amount of results on the peak algebras
can be deduced from the case q = 1 of a qidentity of [11]. Specializing q to other
roots of unity, Krob and the third author introduced and studied higher order peak
