Geodesic
Shifted Tableau Crystals
Séminaire lotharingien de combinatoire, 80B (2018)

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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. 

@article{SLC_2018_80B_a70,
     author = {Maria Gillespie and Jake Levinson, and Kevin Purbhoo},
     title = {Shifted {Tableau} {Crystals}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a70/}
}
TY  - JOUR
AU  - Maria Gillespie
AU  - Jake Levinson,
AU  - Kevin Purbhoo
TI  - Shifted Tableau Crystals
JO  - Séminaire lotharingien de combinatoire
PY  - 2018
VL  - 80B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a70/
ID  - SLC_2018_80B_a70
ER  - 
%0 Journal Article
%A Maria Gillespie
%A Jake Levinson,
%A Kevin Purbhoo
%T Shifted Tableau Crystals
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a70/
%F SLC_2018_80B_a70
Maria Gillespie; Jake Levinson,; Kevin Purbhoo. Shifted Tableau Crystals. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a70/