Prime power variations of higher $Lie_n$ modules

Abstract

We define, for each subset $S$ of the set $\mathcal{P}$ of primes, an $S_n$-module $Lie_n^S$ with interesting properties. $Lie_n^\emptyset$ is the well-known representation $Lie_n$ of $S_n$ afforded by the free Lie algebra, while $Lie_n^\mathcal{P}$ is the module $C\!onj_n$ of the conjugacy action of $S_n$ on $n$-cycles. For arbitrary $S$ the module $Lie_n^{S}$ interpolates between the representations $Lie_n$ and $C\!onj_n.$ We consider the symmetric and exterior powers of $Lie_n^S.$ These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher $Lie_n^S$ 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 $f_n^T$ of homogeneous degree $n.$ We show that the series $\sum_{\lambda, \lambda_i\in T} p_\lambda$ can be expressed as symmetrised powers of the functions $f_n^T$, 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 $f_n^T$, the conjugacy action and the Lie representation.