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arXiv · 1710.05652

Asymptotics for skew standard Young tableaux via bounds for characters

Jehanne Dousse·Valentin Féray
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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.

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Jehanne Dousse, Valentin Féray. 2017-10-16. Asymptotics for skew standard Young tableaux via bounds for characters. https://arxiv.org/abs/1710.05652

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