# Asymptotics for skew standard Young tableaux via bounds for characters

Dousse, Jehanne; Féray, Valentin (2019). Asymptotics for skew standard Young tableaux via bounds for characters. American Mathematical Society. Proceedings, 147(10):4189-4203.

## 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.

## 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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > General Mathematics Physical Sciences > Applied Mathematics Applied Mathematics, General Mathematics English 9 May 2019 16 Dec 2019 09:03 29 Jul 2020 11:29 American Mathematical Society 1088-6826 First published in Proceedings of the American Mathematical Society in 147 (2019), published by the American Mathematical Society Green Publisher DOI. An embargo period may apply. https://doi.org/10.1090/proc/14558