A crystal-like structure on shifted tableaux

Certain investigations in geometry led Kevin Purbhoo, Maria Gillespie and me to define and study new raising and lowering operators on shifted semistandard tableaux. These operators are coplactic for shifted jeu de taquin, giving rise to a crystal-like structure on such tableaux. From it we recover the combinatorics of skew Schur Q functions and the Type B Littlewood-Richardson rule.

 

I'll describe this structure, and say what we know (and don't know) about it. In particular, while we can interpret our operators geometrically, we do not know how to obtain them from representation theory.