Prime power variations of higher $Lie_n$ modules

TL;DR

This paper introduces a family of $S_n$-modules $Lie_n^S$ that interpolate between known representations, explores their symmetric and exterior powers, and demonstrates their Schur positivity through elegant Frobenius characteristic formulas.

Contribution

It defines new modules $Lie_n^S$ for subsets of primes, analyzes their symmetric powers, and establishes Schur positivity of related sums of power sums, unifying and extending prior results.

Findings

Frobenius characteristic expressed as multiplicity-free sum of power sums

Schur positivity of new classes of sums of power sums established

Unified framework for previous and new results on Schur positivity

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.