Prime power variations of higher Lien modules

Abstract

We define, for each subset S of the set P of primes, an Sn-module LienS with interesting properties. Lien∅ is the well-known representation Lien of Sn afforded by the free Lie algebra, while LienP is the module Conjn of the conjugacy action of Sn on n-cycles. For arbitrary S the module LienS interpolates between the representations Lien and Conjn. We consider the symmetric and exterior powers of LienS. These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher LienS modules can be elegantly expressed as a multiplicity-free sum of power sums. In particular this establishes the Schur positivity of new classes of sums of power sums.More generally, for each nonempty subset T of positive integers we define a sequence of symmetric functions fnT of homogeneous degree n. We show that the series ∑λ,λi∈Tpλ can be expressed as symmetrised powers of the functions fnT, analogous to the higher Lie modules first defined by Thrall. This in turn allows us to unify previous results on the Schur positivity of multiplicity-free sums of power sums, as well as investigate new ones. We also uncover some curious plethystic relationships between fnT, the conjugacy action and the Lie representation.