The conjugacy action of $S_n$ and modules induced from centralisers

TL;DR

This paper investigates the conjugacy action of the symmetric group, establishing positivity properties of character table row sums, and introduces a framework for analyzing related induced modules with elegant symmetric function descriptions.

Contribution

It introduces a general framework for analyzing representations induced from centralizers in symmetric groups, revealing positivity properties and symmetric function descriptions of these modules.

Findings

Positivity of row sums of character tables for certain conjugacy class subsets.

Descriptions of induced modules as multiplicity-free sums of power-sum symmetric functions.

Existence of alternative dimension $n!$ representations containing all irreducibles.

Abstract

We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all parts odd. More generally, we study representations related to the conjugacy action of the symmetric group. These arise as sums of submodules induced from centraliser subgroups, and their Frobenius characteristics have elegant descriptions, often as a multiplicity-free sum of power-sum symmetric functions. We describe a general framework in which such representations, and consequently such linear combinations of power-sums, can be analysed. The conjugacy action for the symmetric group, and more generally for a large class of groups, is known to contain every irreducible. We find other representations of dimension $n!$ with this property, including a twisted analogue of the conjugacy action.