New results on the peak algebra

Abstract

The peak algebra $$\mathfrak{P}_{n}$$ 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 $$\mathfrak{P}_{n}$$ . 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 $$\mathfrak{P}_{n}$$ and to characterize the elements of $$\mathfrak{P}_{n}$$ in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals $$\mathfrak{P}_{n}^{j}$$ of $$\mathfrak{P}_{n}$$ , j = 0,..., $${\lfloor \frac{n}{2}\rfloor}$$ , such that $$\mathfrak{P}_{n}^{0}$$ is the linear span of sums of permutations with a common set of interior peaks and $$\smash{\mathfrak{P}_{n}{\lfloor \frac{n}{2}\rfloor}}$$ is the peak algebra. We extend the above results to $$\mathfrak{P}_{n}^{j}$$ , generalizing results of Schocker (the case j = 0).