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


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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  1. Prakash Belkale and Shrawan Kumar. Eigenvalue problem and a new product in cohomology of flag varieties. Inventiones mathematicae, 166(1):185–228, 2006.

    MathSciNet  Article  Google Scholar 

  2. Prakash Belkale, Shrawan Kumar, and Nicolas Ressayre. A generalization of Fulton’s conjecture for arbitrary groups. Math. Ann., 354(2):401–425, 2012.

    MathSciNet  Article  Google Scholar 

  3. Georgia Benkart, Frank Sottile, and Jeffrey Stroomer. Tableau switching: algorithms and applications. J. Combin. Theory Ser. A, 76(1):11–43, 1996.

    MathSciNet  Article  Google Scholar 

  4. Y. M. Chen, A. M. Garsia, and J. Remmel. Algorithms for plethysm. In Combinatorics and algebra (Boulder, Colo., 1983), volume 34 of Contemp. Math., pages 109–153. Amer. Math. Soc., Providence, RI, 1984.

  5. Soojin Cho. A new littlewood-richardson rule for schur functions. Transactions of the American Mathematical Society, 365(2):939–972, 2013.

    MathSciNet  Article  Google Scholar 

  6. Seung-Il Choi and Jae-Hoon Kwon. Crystals and Schur \(P\)-positive expansions. Electron. J. Combin., 25(3):Paper No. 3.7, 27, 2018.

  7. Seung-Il Choi, Sun-Young Nam, and Young-Tak Oh. Bijections among combinatorial models for shifted Littlewood-Richardson coefficients. J. Combin. Theory Ser. A, 128:56–83, 2014.

    MathSciNet  Article  Google Scholar 

  8. William Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry.

  9. Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics.

  10. Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim. Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux. Trans. Amer. Math. Soc., 366(1):457–489, 2014.

  11. Mark D. Haiman. On mixed insertion, symmetry, and shifted Young tableaux. J. Combin. Theory Ser. A, 50(2):196–225, 1989.

    MathSciNet  Article  Google Scholar 

  12. Ricky Ini Liu. An algorithmic Littlewood-Richardson rule. J. Algebraic Combin., 31(2):253–266, 2010.

    MathSciNet  Article  Google Scholar 

  13. Ian Grant Macdonald. Symmetric functions and Hall polynomials. Oxford university press, 1998.

  14. Piotr Pragacz. Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials. In Topics in invariant theory (Paris, 1989/1990), volume 1478 of Lecture Notes in Math., pages 130–191. Springer, Berlin, 1991.

  15. Piotr Pragacz. Addendum: “A generalization of the Macdonald-You formula” [J. Algebra 204 (1998), no. 2, 573–587; MR1624487 (99g:05181)]. J. Algebra, 226(1):639–648, 2000.

  16. Nicolas Ressayre. A cohomology-free description of eigencones in types A, B, and C. Int. Math. Res. Not. IMRN, pages 4966–5005, 2012.

  17. N. Ressayre. Private communication, 2019.

  18. J. B. Remmel and R. Whitney. Multiplying Schur functions. J. Algorithms, 5(4):471–487, 1984.

    MathSciNet  Article  Google Scholar 

  19. Bruce E. Sagan. Shifted tableaux, Schur \(Q\)-functions, and a conjecture of R. Stanley. J. Combin. Theory Ser. A, 45(1):62–103, 1987.

    MathSciNet  Article  Google Scholar 

  20. Luis Serrano. The shifted plactic monoid. Math. Z., 266(2):363–392, 2010.

    MathSciNet  Article  Google Scholar 

  21. Mark Shimozono. Multiplying Schur \(Q\)-functions. J. Combin. Theory Ser. A, 87(1):198–232, 1999.

    MathSciNet  Article  Google Scholar 

  22. John R. Stembridge. Shifted tableaux and the projective representations of symmetric groups. Adv. Math., 74(1):87–134, 1989.

    MathSciNet  Article  Google Scholar 

  23. Ravi Vakil. A geometric Littlewood-Richardson rule. Ann. of Math. (2), 164(2):371–421, 2006. Appendix A written with A. Knutson.

  24. Dennis E. White. Some connections between the Littlewood-Richardson rule and the construction of Schensted. J. Combin. Theory Ser. A, 30(3):237–247, 1981.

    MathSciNet  Article  Google Scholar 

  25. Dale Raymond Worley. A theory of shifted Young tableaux. ProQuest LLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)–Massachusetts Institute of Technology.

  26. A. V. Zelevinsky. A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence. J. Algebra, 69(1):82–94, 1981.

    MathSciNet  Article  Google Scholar 

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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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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).

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  • 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