Crystals for shifted key polynomials

Abstract

This article continues our study of $P$- and $Q$-key polynomials, which are (non-symmetric) "partial" Schur $P$- and $Q$-functions as well as "shifted" versions of key polynomials. Our main results provide a crystal interpretation of $P$- and $Q$-key polynomials, namely, as the characters of certain connected subcrystals of normal crystals associated to the queer Lie superalgebra $\mathfrak{q}_n$. In the $P$-key case, the ambient normal crystals are the $\mathfrak{q}_n$-crystals studied by Grantcharov et al., while in the $Q$-key case, these are replaced by the extended $\mathfrak{q}_n$-crystals recently introduced by the first author and Tong. Using these constructions, we propose a crystal-theoretic lift of several conjectures about the decomposition of involution Schubert polynomials into $P$- and $Q$-key polynomials. We verify these generalized conjectures in a few special cases. Along the way, we establish some miscellaneous results about normal $\mathfrak{q}_n$-crystals and Demazure $\mathfrak{gl}_n$-crystals.