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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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Applied Mathematics, General Mathematics
Language:English
Date:9 May 2019
Deposited On:16 Dec 2019 09:03
Last Modified:16 Dec 2019 09:12
Publisher:American Mathematical Society
ISSN:1088-6826
Additional Information:First published in Proceedings of the American Mathematical Society in 147 (2019), published by the American Mathematical Society
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1090/proc/14558

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