Dimensions of the irreducible representations of the symmetric and alternating group

TL;DR

This paper proves a conjecture by Tong-Viet by showing that certain irreducible representations of the symmetric and alternating groups have unique dimensions not shared between the two groups, using number theory results.

Contribution

It establishes the existence of irreducible representations with unique dimensions for symmetric and alternating groups, confirming Tong-Viet's conjecture.

Findings

Existence of irreducible representations with unique dimensions for A_n and S_n

Proof of Tong-Viet's conjecture

Use of large prime factors in short intervals in the proof

Abstract

We establish the existence of an irreducible representation of $A_n$ whose dimension does not occur as the dimension of an irreducible representation of $S_n$, and vice versa. This proves a conjecture by Tong-Viet. The main ingredient in the proof is a result on large prime factors in short intervals.