Highest weight crystals for Schur Q-functions

TL;DR

This paper introduces a modified category of crystals for the queer Lie superalgebra $\

Contribution

It develops a new category of $\

Findings

Crystals have characters as Schur $Q$-polynomials.

Existence of an extra crystal operator and a different tensor product.

Action of the hyperoctahedral group exchanging highest and lowest weights.

Abstract

Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra $\mathfrak{q}_n$. Such $\mathfrak{q}_n$-crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur $P$-polynomials. This article studies a modified form of this category, whose connected normal objects again have unique highest weight elements but now possess characters that are Schur $Q$-polynomials. The crystals in this category have some interesting features not present for ordinary $\mathfrak{q}_n$-crystals. For example, there is an extra crystal operator, a different tensor product, and an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of $\mathfrak{q}_n$-crystal structures on certain families of shifted tableaux and factorized reduced words. We describe extended forms of these structures that give similar examples in our new category.