Séminaire Lotharingien de Combinatoire, 80B.71 (2018), 12 pp.

Maria Gillespie, Jake Levinson, and Kevin Purbhoo

Shifted Tableau Crystals

Abstract. We introduce coplactic raising and lowering operators E'i, F'i, Ei, and Fi 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 give local axioms for these crystals, which closely resemble the Stembridge axioms for type A. Finally, we give a new criterion for such tableaux to be ballot.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

The following versions are available: