# Crystal graphs for shifted tableaux

## TL;DR

This paper introduces crystal operators on semistandard shifted tableaux, providing new proofs of Schur P-function positivity and constructing a queer crystal, along with a rectification map linking shifted and Young tableaux.

## Contribution

It defines new crystal operators and a queer crystal structure on shifted tableaux, offering novel proofs of Schur P-function positivity and a dual shifted insertion algorithm.

## Key findings

- Established crystal operators on shifted tableaux
- Proved Schur P-functions are Schur positive using new methods
- Constructed a queer crystal and a rectification map

## Abstract

We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur $P$-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a given shape, providing a new proof that products of Schur $P$-functions are Schur $P$-positive. We also give a rectification map from shifted tableaux to Young tableaux that commutes with the crystal operators and provides a dual algorithm to shifted insertion.