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Asymptotics for skew standard Young tableaux via bounds for characters


Authors: Jehanne Dousse and Valentin Féray
Journal: Proc. Amer. Math. Soc. 147 (2019), 4189-4203
MSC (2010): Primary 05A16, 05E05, 05E10
DOI: https://doi.org/10.1090/proc/14558
Published electronically: May 9, 2019
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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 $ \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 Information

Jehanne Dousse
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8032 Zürich, Switzerland
Address at time of publication: CNRS, Universite Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, F-69622 Villeurbanne, France
Email: dousse@math.cnrs.fr

Valentin Féray
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8032 Zürich, Switzerland
Email: valentin.feray@math.uzh.ch

DOI: https://doi.org/10.1090/proc/14558
Keywords: Skew shapes, standard Young tableaux, asymptotics, characters
Received by editor(s): January 31, 2018
Received by editor(s) in revised form: January 18, 2019
Published electronically: May 9, 2019
Additional Notes: Both authors were partially supported by grant SNF-149461 from the Swiss National Science Foundation.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2019 American Mathematical Society