Bounds on skew dimensions and characters of symmetric groups via thick hook decompositions

TL;DR

This paper establishes new bounds on the number of standard tableaux of skew shapes and symmetric group characters using thick hook decompositions, improving existing bounds for large support permutations.

Contribution

It introduces a novel approach combining thick hook decompositions with elementary counting to bound symmetric group characters, enhancing previous results.

Findings

Bound on skew shape tableaux via thick hook decompositions

Uniform bound on symmetric group characters

Improved bounds for permutations with large support

Abstract

We bound the number of standard tableaux of skew shapes via thick hook decompositions in the Naruse hook length formula. Combining this with elementary counting arguments in the Murnaghan--Nakayama rule, we establish a uniform bound on characters of symmetric groups $\mathfrak{S}_n$. In the case of balanced representations, this improves on the character bounds of F\'eray and \'Sniady for permutations with support size at least $n^{2/3}$, and is sharp for permutations with support size of order $n$.