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UNITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS MARCELO AGUIAR, JEAN-CHRISTOPHE NOVELLI, JEAN-YVES THIBON
 

Summary: UNITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS
MARCELO AGUIAR, JEAN-CHRISTOPHE NOVELLI, JEAN-YVES 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 Mantaci-Reutenauer 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 q-identity of [11]. Specializing q to other
roots of unity, Krob and the third author introduced and studied higher order peak

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics