# Restriction of characters to subgroups of wreath products and basic sets for the symmetric group

## TL;DR

This paper provides a detailed decomposition of irreducible characters of certain wreath products into irreducible components, connecting these to the structure of symmetric groups and their basic sets, with explicit formulas involving Littlewood-Richardson coefficients.

## Contribution

It offers a new explicit decomposition formula for restrictions of irreducible characters to wreath products, enhancing understanding of symmetric group characters and their basic sets.

## Key findings

- Decomposition of characters into irreducibles using Littlewood-Richardson coefficients
- Connection between wreath product restrictions and symmetric group basic sets
- Explicit formulas for p-regular element restrictions

## Abstract

In this paper, we give the decomposition into irreducible characters of the restriction to the wreath product $\mathbb{Z}_{p-1} \wr \mathfrak{S}_w$ of any irreducible character of $(\mathbb{Z}_p \rtimes \mathbb{Z}_{p-1}) \wr \mathfrak{S}_w$, where $p$ is any odd prime, $w \geq 0$ is an integer, and $\mathbb{Z}_p$ and $\mathbb{Z}_{p-1}$ denote the cyclic groups of order $p$ and $p-1$ respectively. This answers the question of how to decompose the restrictions to $p$-regular elements of irreducible characters of the symmetric group $\mathfrak{S}_n$ in the $\mathbb{Z}$-basis corresponding to the $p$-basic set of $\mathfrak{S}_n$ described by Brunat and Gramain in [1]. The result is given in terms of the Littlewood-Richardson coefficients for the symmetric group.