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 $n$.