Peak algebra

In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group Sn, studied by. It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation σ on {1,2,...,n} is an index i such that σ(i–1)<σ(i)>σ(i+1).) It is a left ideal of the descent algebra. The direct sum of the peak algebras for all n has a natural structure of a Hopf algebra.