ALGEBRAIC COMBINATORICS

A crystal-like structure on shifted tableaux
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 693-725.

We introduce coplactic raising and lowering operators ${E}_{i}^{\prime }$, ${F}_{i}^{\prime }$, ${E}_{i}$, and ${F}_{i}$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of “doubled crystal” structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood–Richardson tableaux, and their generating functions are the (skew) Schur $Q$-functions. We also give a new criterion for such tableaux to be ballot.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.110
Classification: 05E99,  05E05
Keywords: Combinatorial crystals, shifted Young tableaux, symmetric function theory, orthogonal Grassmannian.
Gillespie, Maria 1; Levinson, Jake 2; Purbhoo, Kevin 3

1 Department of Mathematics Colorado State University Fort Collins, CO, USA
2 Department of Mathematics University of Washington Seattle, WA, USA
3 Department of Combinatorics and Optimization University of Waterloo Waterloo, ON, Canada
@article{ALCO_2020__3_3_693_0,
author = {Gillespie, Maria and Levinson, Jake and Purbhoo, Kevin},
title = {A crystal-like structure on shifted tableaux},
journal = {Algebraic Combinatorics},
pages = {693--725},
publisher = {MathOA foundation},
volume = {3},
number = {3},
year = {2020},
doi = {10.5802/alco.110},
mrnumber = {4113603},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.110/}
}
TY  - JOUR
TI  - A crystal-like structure on shifted tableaux
JO  - Algebraic Combinatorics
PY  - 2020
DA  - 2020///
SP  - 693
EP  - 725
VL  - 3
IS  - 3
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.110/
UR  - https://www.ams.org/mathscinet-getitem?mr=4113603
UR  - https://doi.org/10.5802/alco.110
DO  - 10.5802/alco.110
LA  - en
ID  - ALCO_2020__3_3_693_0
ER  - 
%0 Journal Article
%T A crystal-like structure on shifted tableaux
%J Algebraic Combinatorics
%D 2020
%P 693-725
%V 3
%N 3
%I MathOA foundation
%U https://doi.org/10.5802/alco.110
%R 10.5802/alco.110
%G en
%F ALCO_2020__3_3_693_0
Gillespie, Maria; Levinson, Jake; Purbhoo, Kevin. A crystal-like structure on shifted tableaux. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 693-725. doi : 10.5802/alco.110. https://alco.centre-mersenne.org/articles/10.5802/alco.110/

[1] Assaf, Sami Shifted dual equivalence and Schur P-positivity, J. Comb., Volume 9 (2018) no. 2, pp. 279-308 | Article | MR: 3763646 | Zbl: 1378.05210

[2] Assaf, Sami; Oguz, Ezgi Kantarcı Toward a Local Characterization of Crystals for the Quantum Queer Superalgebra, Ann. Comb., Volume 24 (2020), pp. 3-46 | Article | MR: 4078137 | Zbl: 07181803

[3] Bump, Daniel; Schilling, Anne Crystal Bases: Representations and Combinatorics, World Scientific, 2017 | Article | Zbl: 06690908

[4] Cho, Soojin A new Littlewood–Richardson rule for Schur P-functions, Trans. Am. Math. Soc., Volume 365 (2013) no. 2, pp. 939-972 | MR: 2995379 | Zbl: 1262.05152

[5] Choi, Seung-Il; Kwon, Jae-Hoon Crystals and Schur P-positive expansions, Electron. J. Comb., Volume 25 (2018) no. 3, Paper no. P3.7, 27 pages | MR: 3829293 | Zbl: 1394.17035

[6] Gillespie, Maria; Hawkes, Graham; Poh, Wencin; Schilling, Anne Characterization of queer supercrystals (2019) (https://arxiv.org/abs/1809.04647)

[7] Gillespie, Maria; Levinson, Jake Monodromy and K-theory of Schubert curves via generalized jeu de taquin, J. Algebr. Comb., Volume 45 (2017), pp. 191-243 | Article | MR: 3591376 | Zbl: 1355.05272

[8] Gillespie, Maria; Levinson, Jake Axioms for Shifted Tableau Crystals, Electron. J. Comb., Volume 26 (2019) no. 2, Paper no. P2.2, 38 pages | MR: 3940779 | Zbl: 1414.05296

[9] Gillespie, Maria; Levinson, Jake; Purbhoo, Kevin Schubert curves in the orthogonal Grassmannian (2019) (https://arxiv.org/abs/1903.01673)

[10] Grantcharov, Dimitar; Jung, Ji Hye; Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux, Trans. Am. Math. Soc., Volume 366 (2014) no. 1, pp. 457-489 | Article | MR: 3118402 | Zbl: 1336.17013

[11] Haiman, Mark D. Dual equivalence with applications, including a conjecture of Proctor, Discrete Math., Volume 99 (1992) no. 1–3, pp. 79-113 | Article | MR: 1158783 | Zbl: 0760.05093

[12] Hiroshima, Toya $𝔮$-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type $B$, Publ. Res. Inst. Math. Sci., Volume 55 (2019) no. 2, pp. 369-399 | Article | MR: 3941487 | Zbl: 1416.05294

[13] Kashiwara, Masaki Crystalizing the q-analogue of universal enveloping algebras, Commun. Math. Phys., Volume 133 (1990) no. 2, pp. 249-260 | Article | MR: 1090425 | Zbl: 0724.17009

[14] Macdonald, Ian G. Symmetric Functions and Hall Polynomials, Oxford Univ. Press, 1979 | Zbl: 0487.20007

[15] Morse, Jennifer; Schilling, Anne Crystal approach to affine Schubert calculus, Int. Math. Res. Not., Volume 2016 (2016) no. 8, pp. 2239-2294 | Article | MR: 3519114 | Zbl: 1404.14057

[16] Pragacz, Piotr Algebro-geometric applications of Schur S- and Q-polynomials, Topics in invariant theory (Lect. Notes Math.), Volume 1478, Springer, 1991, pp. 130-191 Séminaire d’Algèbre Dubreil–Malliavin 1989–1990 (M.-P. Malliavin ed.) | Article | MR: 1180989 | Zbl: 0783.14031

[17] Sagan, Bruce E. Shifted tableaux, Schur Q-functions and a conjecture of R. Stanley, J. Comb. Theory, Ser. A, Volume 45 (1987), pp. 62-103 | Article | MR: 883894 | Zbl: 0661.05010

[18] Sagan, Bruce E. The Symmetric Group, Springer, New York, 2001 | Article | Zbl: 0964.05070

[19] Sage Developers SageMath, the Sage Mathematics Software System (Version 7.6), 2017 (https://www.sagemath.org)

[20] Schur, Issai Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., Volume 139 (1911), pp. 155-250 | Zbl: 42.0154.02

[21] Serrano, Luis The shifted plactic monoid, Math. Z., Volume 266 (2010), pp. 363-392 | Article | MR: 2678632 | Zbl: 1231.05279

[22] Shimozono, Mark Multiplying Schur Q-functions, J. Comb. Theory, Ser. A, Volume 87 (1999) no. 1, pp. 198-232 | Article | MR: 1698253 | Zbl: 0978.05073

[23] Stembridge, John R. Shifted tableaux and the projective representations of the symmetric group, Adv. Math., Volume 74 (1989), pp. 87-134 | Article | MR: 991411 | Zbl: 0677.20012

[24] Stembridge, John R. A local characterization of simply-laced crystals, Trans. Am. Math. Soc., Volume 355 (2003) no. 12, pp. 4807-4823 | Article | MR: 1997585 | Zbl: 1047.17007

[25] van Leeuwen, Mark A. A. The Littlewood–Richardson Rule and related combinatorics, MSJ Mem., Volume 11 (2001), pp. 95-145 | MR: 1862150 | Zbl: 0991.05101

[26] Worley, Dale R. A theory of shifted Young tableau (1984) (Ph. D. Thesis) | MR: 2941073

Cited by Sources: