On low-degree representations of the symmetric group

TL;DR

This paper classifies all irreducible modular representations of the symmetric group with dimension at most n^3, providing dimension formulas and leveraging theoretical and computational methods.

Contribution

It improves existing dimension bounds and offers a comprehensive classification of low-degree irreducible representations of symmetric groups.

Findings

Complete classification of irreducible modular representations with dimension ≤ n^3.

Derived explicit dimension formulas for these representations.

Enhanced understanding of decomposition numbers and their role in representation theory.

Abstract

The aim of the present paper is to obtain a classification of all the irreducible modular representations of the symmetric group on $n$ letters of dimension at most $n^3$, including dimension formulae. This is achieved by improving an idea, originally due to G. James, to get hands on dimension bounds, by building on the current knowledge about decomposition numbers of symmetric groups and their associated Iwahori-Hecke algebras, and by employing a mixture of theory and computation.