The peak algebra and the descent algebras of types B and D

Abstract

Abstract. We show the existence of a unital subalgebra Pn of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that Pn is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra Pn contains a two-sided ideal • P n which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form 0 → • P n → Pn → P n−2 → 0. We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions n and n 2 +1) by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces Pn and • P n over n ≥ 0 and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.

Coloured peak algebras and Hopf algebras

“…Here, our approach is to study properties of equivalence relations on G = n≥0 G n and to determine when they give rise to algebraic structures. Many of the algebraic objects obtained in this way are G-colouring of the structures in the literature, such as the peak algebras [1,9,21] or the Loday-Ronco Hopf algebra of trees [17]. Our work is the unifying generalization of a series of results starting in [5] and continuing in [6,14,26].…”

“…These objects are particularly interesting as their structure constants may encode various invariants in geometry, in physics or in computer science. This is particularly true in [1,9,12,13,17,21,24,29], just to cite a few, and in [2] we find the beginning of a general theory of combinatorial Hopf algebras. Most of the algebraic structures under study are subalgebras, subcoalgebras, Hopf subalgebras or quotients of k[S] = n≥0 kS n , where k is a field (of characteristic 0) and kS n is the group algebra of the symmetric group S n .…”

“…It is then natural to ask if the multitude of substructures and quotients of k[S] studied previously can be G-coloured. A few answer have been given, see for instance [1,6,22,15]. In general this requires some work, and is not possible in all cases.…”

Some Combinatorial Arrays Related to the Lotka-Volterra System

“…In this paper, we explore some context-free grammars suggested by (1). In fact, there are many other extension of (1).…”

“…The number of peaks in a permutation is an important combinatorial statistic. See, e.g., [1,5,7,10] and the references therein. However, the question of whether the first and/or last entry may qualify as a peak (or valley) gives rise to several different definitions.…”