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

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 $n$.