New Results on the Peak Algebra

The peak algebra Pn is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of Pn. We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of Pn and to characterize the elements of Pn in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals Pn of Pn, j = 0, . . . , ⌊ n 2 ⌋, such that P0n is the linear span of sums of permutations with a common set of interior peaks and P ⌊n 2 ⌋ n is the peak algebra. We extend the above results to Pn, generalizing results of Schocker (the case j = 0). Introduction A descent of a permutation σ ∈ Sn is a position i for which σ(i) > σ(i + 1), while a peak is a position i for which σ(i− 1) < σ(i) > σ(i+ 1). One aspect of the algebraic theory of peaks was initiated by Stembridge [21], another by Nyman [14]. The peak algebra Pn was introduced in [1]. It is a unital subalgebra of the group algebra of the symmetric group Sn, obtained as the linear span of sums of permutations with a common set of peaks. The construction is analogous to that of the descent algebra of Sn, denoted Sol(An−1), which is obtained as the linear span of sums of permutations with a common set of descents. Pn is a subalgebra of Sol(An−1). The descent algebra has been the object of numerous works; for a recent survey see [17]. The peak algebra, or closely related objects, has been studied in [1, 5, 8, 16], from different perspectives. The descent algebra construction, due to Solomon, can be extended to all finite Coxeter groups [19]. Let Bn be the group of signed permutations: Bn = Sn ⋉ Z n 2 , and φ : Bn → Sn the canonical projection (the map that forgets the signs). A basic observation of [1] is that this map sends the descent algebra of Bn, denoted Sol(Bn), onto the peak algebra Pn. This allows us to derive properties of the peak algebra from known properties of the descent algebra of Bn. This point of view is emphasized again in this work. Date: June 9, 2004. 2000 Mathematics Subject Classification. Primary 05E99, 20F55; Secondary: 05A99, 16W30.