Asymptotics for skew standard Young tableaux via bounds for characters

TL;DR

This paper investigates the asymptotic behavior of the number of skew standard Young tableaux for balanced diagrams, providing expansions, bounds, and conjectures across different growth regimes of the skew shape.

Contribution

It introduces new asymptotic expansions and bounds for the count of skew standard Young tableaux using character theory of symmetric groups.

Findings

Asymptotic expansion for small skew shapes when ||=o(||^{1/3})

Sharp upper bounds for intermediate growth regimes when ||=o(||^{1/2})

Conjectures on the order of magnitude for larger skew shapes

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 bigger $|\mu|$, 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.