The peak algebra of the symmetric group revisited

TL;DR

This paper revisits the peak algebra of the symmetric group, providing new combinatorial and algebraic characterizations, and explores its structure as a Hopf sub-algebra, including radical, modules, and primitive Lie algebra properties.

Contribution

It offers a self-contained analysis of the peak algebra, establishing new characterizations and structural properties, and connecting it to Lie theory and Hopf algebra frameworks.

Findings

Characterization of P_n via combinatorics and algebra

Description of the Jacobson radical and modules of P_n

Proof that the primitive Lie algebra of P is free

Abstract

The linear span P_n of the sums of all permutations in the symmetric group S_n with a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra D_n; and the direct sum P of all P_n is a Hopf sub-algebra of the direct sum D of all D_n, dual to the Stembridge algebra of peak functions. In our self-contained approach, peak counterparts of several results on the descent algebra are established, including a simple combinatorial characterization of the algebra P_n; an algebraic characterization of P_n based on the action on the Poincar'e-Birkhoff-Witt basis of the free associative algebra; the display of peak variants of the classical Lie idempotents; an Eulerian-type sub-algebra of P_n; a description of the Jacobson radical of P_n and its nil-potency index, of the principal indecomposable and irreducible P_n-modules, and of the Cartan matrix of P_n. Furthermore, it is shown that the primitive Lie algebra of P is free, and that P is its enveloping algebra.