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no. 3
A crystal-like structure on shifted tableaux
Gillespie, Maria1; Levinson, Jake2; Purbhoo, Kevin3
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
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.

Received:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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