Crystals for shifted key polynomials

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.