# 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 ${E}_{i}^{\text{'}}$, ${F}_{i}^{\text{'}}$, ${E}_{i}$, 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: 2020-01-13
Accepted: 2020-01-14
Published online: 2020-06-02
DOI: https://doi.org/10.5802/alco.110
Classification: 05E99,  05E05
Keywords: Combinatorial crystals, shifted Young tableaux, symmetric function theory, orthogonal Grassmannian.
@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},
publisher = {MathOA foundation},
volume = {3},
number = {3},
year = {2020},
pages = {693-725},
doi = {10.5802/alco.110},
language = {en},
url = {alco.centre-mersenne.org/item/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/item/ALCO_2020__3_3_693_0/

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