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 , F i , 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.

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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
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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/

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