Decompositions of some twisted Foulkes characters

TL;DR

This paper provides explicit decompositions of twisted Foulkes characters, specifically for cases where the partition has two rows, two columns, or is a hook, enhancing understanding of their structure.

Contribution

It offers new explicit decompositions of twisted Foulkes characters for specific partition shapes, advancing the combinatorial understanding of plethysm.

Findings

Decomposition formulas for two-row partitions

Decomposition formulas for two-column partitions

Decomposition formulas for hook partitions

Abstract

We decompose the twisted Foulkes characters $\phi^{(2^n)}_\nu$, or equivalently the plethysm $s_\nu \circ s_{(2)}$, in the cases where $\nu$ has either two rows or two columns, or is a hook partition.