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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Asymptotics for skew standard Young tableaux via bounds for characters
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by Jehanne Dousse and Valentin Féray PDF
Proc. Amer. Math. Soc. 147 (2019), 4189-4203 Request permission

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 larger $|\mu |$, 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.

References
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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
  • MR Author ID: 1036858
  • ORCID: 0000-0001-6825-0389
  • 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
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4189-4203
  • MSC (2010): Primary 05A16, 05E05, 05E10
  • DOI: https://doi.org/10.1090/proc/14558
  • MathSciNet review: 4002535