New results on the peak algebra

Abstract

The peak algebra ℬn 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 ℬn. 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 ℬn and to characterize the elements of ℬn in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals ℬnj, of ℬn, j = 0,..., ⌊n/2⌋, such that ℬn0 is the linear span of sums of permutations with a common set of interior peaks and ℬn⌊n/2⌋ is the peak algebra. We extend the above results to ℬnj, generalizing results of Schocker (the case j = 0). © Springer Science + Business Media, Inc. 2006.