Asymptotics for skew standard Young tableaux via bounds for characters

Abstract

We are interested in the asymptotics of the number of standard Young tableaux$ f^{\lambda /\mu }$ of a given skew shape λ/μ. We mainly restrict ourselves to the case where both diagrams are balanced, but investigate all growth regimes of$ \vert\mu \vert$ compared to $ \vert\lambda \vert$, from $ \vert\mu \vert$ fixed to $ \vert\mu \vert$ of order $ \vert\lambda \vert$. When $ \vert\mu \vert=o(\vert\lambda \vert^{1/3})$, we get an asymptotic expansion to any order. When $ \vert\mu \vert=o(\vert\lambda \vert^{1/2})$, we get a sharp upper bound. For larger $ \vert\mu \vert$, we prove a weaker bound and give a conjecture on what we believe to be the correct order of magnitude.
Our results are obtained by expressing $ f^{\lambda /\mu }$ in terms of irreducible character values of the symmetric group and applying known upper bounds on characters.