The Peak Algebra of the Symmetric Group

The peak set of a permutation σ is the set {i : σ(i − 1) < σ(i) > σ(i + 1)}. The group algebra of the symmetric group Sn admits a subalgebra in which elements are sums of permutations with a common descent set. In this paper we show the existence of a subalgebra of this descent algebra in which elements are sums of permutations sharing a common peak set. To prove the existence of this peak algebra we use the theory of enriched (P, γ)-partitions and the algebra of quasisymmetric peak functions studied by Stembridge (Trans. Amer. Math. Soc. 349 (1997) 763–788).