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 $\lambda /\mu$. We mainly restrict ourselves to the case where both diagrams are balanced, but investigate all growth regimes of $|\mu |$ compared to $|\lambda |$, from $|\mu |$ fixed to $|\mu |$ of order $|\lambda |$. When $|\mu |=o(|\lambda |^{1/3})$, we get an asymptotic expansion to any order. When $|\mu |=o(|\lambda |^{1/2})$, we get a sharp upper bound. For larger $|\mu |$, 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.