On the local representation theory of symmetric groups

TL;DR

This paper investigates the local representation theory of symmetric groups by analyzing the action of the normalizer of Sylow p-subgroups on their irreducible characters, establishing a correspondence with certain functions.

Contribution

It introduces a new explicit correspondence between irreducible characters of Sylow p-subgroups and equivalence classes of functions, illuminating their Galois action.

Findings

Describes the normalizer action on irreducible characters

Establishes a bijection with explicitly defined functions

Provides insights into Galois actions on characters

Abstract

Given a Sylow $p$-subgroup $P$ of a symmetric group, we describe the action of its normalizer on $\mathrm{Irr}(P)$. To this end, we establish a one-to-one correspondence between the irreducible characters of $P$ and certain equivalence classes of explicitly defined functions, which are also naturally suited to describing the Galois action.