A crystal-like structure on shifted tableaux

TL;DR

This paper introduces new crystal operators on shifted tableaux that form type A Kashiwara crystals and combine to reveal a doubled crystal structure, connecting to type B Schubert calculus and Schur Q functions.

Contribution

It presents coplactic raising and lowering operators on shifted tableaux that form a novel doubled crystal structure, linking combinatorics of shifted tableaux to type B Schubert calculus.

Findings

Primed and unprimed operators form type A Kashiwara crystals

The doubled crystal structure recovers shifted Littlewood-Richardson tableaux

Generating functions are Schur Q functions

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.