Axioms for shifted tableau crystals

TL;DR

This paper introduces local axioms that uniquely define the crystal structure on shifted tableaux, providing a new approach to understanding Schur Q-positive expansions in symmetric functions.

Contribution

It develops a novel set of axioms for shifted tableau crystals, paralleling Stembridge's axioms for type A, enabling new proofs and insights in symmetric function theory.

Findings

Axioms uniquely characterize shifted tableau crystals

New method for proving Schur Q-positivity

Enhanced understanding of symmetric function expansions

Abstract

We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed in a previous paper by Gillespie, Levinson, and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur $Q$-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.