Plethysm and a character embedding problem of Miller

TL;DR

This paper applies a plethystic formula of Littlewood to address a character embedding problem of Miller and reestablishes a related character congruence result.

Contribution

It introduces a novel application of plethystic formulas to solve a specific character embedding problem and reprove a known congruence result.

Findings

Solved Miller's embedding question using plethystic formulas

Reproved Miller's character congruence result

Connected symmetric group characters with plethystic techniques

Abstract

We use a plethystic formula of Littlewood to answer a question of Miller on embeddings of symmetric group characters. We also reprove a result of Miller on character congruences.