Geodesic
A crystal-like structure on shifted tableaux
Gillespie, Maria1; Levinson, Jake2; Purbhoo, Kevin3
1Department of Mathematics Colorado State University Fort Collins, CO, USA
2Department of Mathematics University of Washington Seattle, WA, USA
3Department of Combinatorics and Optimization University of Waterloo Waterloo, ON, Canada
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 693-725

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MR

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
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
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