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On the Shifted Littlewood–Richardson Coefficients and the Littlewood–Richardson Coefficients

Abstract

We give a new interpretation of the shifted Littlewood–Richardson coefficients \(f_{\lambda \mu }^\nu \) (\(\lambda ,\mu ,\nu \) are strict partitions). The coefficients \(g_{\lambda \mu }\) which appear in the decomposition of Schur Q-function \(Q_\lambda \) into the sum of Schur functions \(Q_\lambda = 2^{l(\lambda )}\sum \nolimits _{\mu }g_{\lambda \mu }s_\mu \) can be considered as a special case of \(f_{\lambda \mu }^\nu \) (here \(\lambda \) is a strict partition of length \(l(\lambda )\)). We also give another description for \(g_{\lambda \mu }\) as the cardinal of a subset of a set that counts Littlewood–Richardson coefficients \(c_{\mu ^t\mu }^{{\tilde{\lambda }}}\). This new point of view allows us to establish connections between \(g_{\lambda \mu }\) and \(c_{\mu ^t \mu }^{{\tilde{\lambda }}}\). More precisely, we prove that \(g_{\lambda \mu }=g_{\lambda \mu ^t}\), and \(g_{\lambda \mu } \le c_{\mu ^t\mu }^{{\tilde{\lambda }}}\). We conjecture that \(g_{\lambda \mu }^2 \le c^{{\tilde{\lambda }}}_{\mu ^t\mu }\) and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.

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Acknowledgements

The author would like to express his sincere gratitude to his supervisors Prof. Nicolas Ressayre and Prof. Kenji Iohara for suggesting the subject and for many useful discussion, inspiring ideas during the work. He is also grateful to their corrections and their teaching of how to write better, understandable and clear. He would also like to thank Prof. Cédric Lecouvey and Prof. Shrawan Kumar - the reporters for his thesis, for reading it carefully, pointing out several important errors that improved the text. The author is grateful to the referees for their comments to improve the manuscript.

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Correspondence to Duc Khanh Nguyen.

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Communicated by Jang Soo Kim

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Appendix

Appendix

The construction of the set \(\widetilde{\mathcal {O}}(\nu /\mu )\) in the Example 4.5

figure l

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Nguyen, D.K. On the Shifted Littlewood–Richardson Coefficients and the Littlewood–Richardson Coefficients. Ann. Comb. 26, 221–260 (2022). https://doi.org/10.1007/s00026-022-00566-7

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Keywords

  • Young tableaux
  • Schur functions
  • Littlewood–Richardson coefficients
  • Tableau switching
  • Grassmannians
  • Schubert varieties
  • Shifted tableaux
  • Schur Q-functions
  • Shifted Littlewood–Richardson coefficients
  • Lagrangian Grassmannians

Mathematics Subject Classification

  • Primary 05E10
  • Secondary 05E05
  • 14M15