Minimal and maximal constituents of twisted Foulkes characters

TL;DR

This paper establishes combinatorial rules to identify the minimal and maximal partitions labeling irreducible constituents of certain symmetric group characters, generalizing Foulkes characters and resolving related conjectures.

Contribution

It introduces new combinatorial rules for minimal and maximal constituents in twisted Foulkes characters and plethysms, extending previous understanding and confirming conjectures.

Findings

Determined all minimal and maximal partitions in plethysms s_ν ∘ s_(m).

Proved two conjectures of Agaoka regarding lexicographically least constituents.

Generalized Foulkes permutation characters to broader symmetric group characters.

Abstract

We prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms s_\nu \circ s_(m). As a corollary we prove two conjectures of Agaoka on the lexicographically least constituents of the plethysms s_\nu \circ s_(m) and s_\nu \circ s_(1^m).