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.
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