Highest weight crystals for Schur Q-functions

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.

arXiv comment55 pages, 2 figures; v2: minor corrections, updated references, added exposition

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