 
Summary: NEW RESULTS ON THE PEAK ALGEBRA
MARCELO AGUIAR, KATHRYN NYMAN, AND ROSA ORELLANA
Abstract. 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 [n1] (and of certain classes of compositions of n
called almostodd 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 semiidempotent
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
Pj
n of Pn, j = 0, . . . , n
2 , such that P0
n 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 Pj
n, generalizing results of Schocker (the case j = 0).
