- Volume 56 Issue 4
We provide two shifted analogues of the tableau switching process due to Benkart, Sottile, and Stroomer; the shifted tableau switching process and the modified shifted tableau switching process. They are performed by applying a sequence of elementary transformations called switches and shares many nice properties with the tableau switching process. For instance, the maps induced from these algorithms are involutive and behave very nicely with respect to the lattice property. We also introduce shifted generalized evacuation which exactly agrees with the shifted J-operation due to Worley when applied to shifted Young tableaux of normal shape. Finally, as an application, we give combinatorial interpretations of Schur P- and Schur Q-function related identities.
shifted tableau switchings;shifted jeu de taquin;Schur P- and Schur Q-functions;shifted Littlewood-Richardson coefficients
- G. Benkart, F. Sottile, and J. Stroomer, Tableau switching: algorithms and applications, J. Combin. Theory Ser. A 76 (1996), no. 1, 11-43. https://doi.org/10.1006/jcta.1996.0086
- A. S. Buch, A. Kresch, and H. Tamvakis, Littlewood-Richardson rules for Grassmannians, Adv. Math. 185 (2004), no. 1, 80-90. https://doi.org/10.1016/S0001-8708(03)00165-8
- S. Cho, A new Littlewood-Richardson rule for Schur P-functions, Trans. Amer. Math. Soc. 365 (2013), no. 2, 939-972.
- S.-I. Choi, S.-Y. Nam, and Y.-T. Oh, Bijections among combinatorial models for shifted Littlewood-Richardson coefficients, J. Combin. Theory Ser. A 128 (2014), 56-83.
- E. A. DeWitt, Identities Relating Schur s-Functions and Q-Functions, ProQuest LLC, Ann Arbor, MI, 2012.
- D. Grantcharov, J. H. Jung, S.-J. Kang, M. Kashiwara, and M. Kim, Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux, Trans. Amer. Math. Soc. 366 (2014), no. 1, 457-489. https://doi.org/10.1090/S0002-9947-2013-05866-7
- M. D. Haiman, On mixed insertion, symmetry, and shifted Young tableaux, J. Combin. Theory Ser. A 50 (1989), no. 2, 196-225.
- P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992.
- G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, MA, 1981.
- O. Pechenik and A. Yong, Genomic tableaux, J. Algebraic Combin. 45 (2017), no. 3, 649-685. https://doi.org/10.1007/s10801-016-0720-8
- B. E. Sagan, Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), no. 1, 62-103.
- M. P. Schutzenberger, Quelques remarques sur une construction de Schensted, Canad. J. Math. 13 (1961), 117-128.
- M. P. Schutzenberger, Promotion des morphismes d'ensembles ordonnes, Discrete Math. 2 (1972), 73-94.
- L. Serrano, The shifted plactic monoid, Math. Z. 266 (2010), no. 2, 363-392. https://doi.org/10.1007/s00209-009-0573-0
- K. Shigechi, Shifted tableaux and product of Schur's symmetric functions, arXiv:1705.06437v1.
J. R. Stembridge, On symmetric functions and the spin characters of
$S_n$, in Topics in algebra, Part 2 (Warsaw, 1988), 433-453, Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990.
- J. R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), no. 1, 87-134.
- H. Thomas and A. Yong, A combinatorial rule for (co)minuscule Schubert calculus, Adv. Math. 222 (2009), no. 2, 596-620. https://doi.org/10.1016/j.aim.2009.05.008
- D. R. Worley, A theory of shifted Young tableaux, ProQuest LLC, Ann Arbor, MI, 1984.
Supported by : NRF, Samsung Science and Technology Foundation