Shifted tableaux crystals

TL;DR

This paper introduces new crystal operators on shifted skew semistandard tableaux, forming a doubled crystal structure that models type B Schubert calculus and Schur Q-functions.

Contribution

It defines coplactic raising and lowering operators that form type A crystals and combines them into a doubled crystal structure related to type B combinatorics.

Findings

Primed and unprimed operators form independent type A Kashiwara crystals.

The doubled crystal structure captures the combinatorics of type B Schubert calculus.

Provides local axioms and a ballot criterion for these tableaux.

Abstract

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 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.