Multiplicity-free representations of symmetric groups

TL;DR

This paper classifies all multiplicity-free permutation characters of symmetric groups of degree 66 or more, providing a comprehensive list of irreducible characters that can appear in such representations and exploring related monomial characters.

Contribution

It offers the first complete classification of multiplicity-free permutation characters for large symmetric groups, extending previous partial results and analyzing implications for Specht filtrations.

Findings

Complete list of multiplicity-free permutation characters for symmetric groups of degree ≥66.

Classification of multiplicity-free irreducible characters in these groups.

Analysis of Specht filtrations in prime characteristic fields.

Abstract

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic. Remark: parallel work of Godsil and Meagher (arXiv:math/0612567) gives an independent classification of the multiplicity-free permutation characters of symmetric groups of all degrees.