\(\mathfrak{q}\)-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type \(B\) (Q2418849)

Summary: Crystal basis theory for the queer Lie superalgebra was developed in [\textit{D. Grantcharov} et al., Trans. Am. Math. Soc. 366, No. 1, 457--489 (2014; Zbl 1336.17013) and J. Eur. Math. Soc. (JEMS) 17, No. 7, 1593--1627 (2015; Zbl 1336.17014)], where it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply \(\mathfrak{q}\)-crystal structure. In this paper, we explore the \(\mathfrak{q}\)-crystal structure of primed tableaux, semistandard marked shifted tableaux and that of signed unimodal factorizations of reduced words of type \(B\). We give explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors. We clarify the relation between signed unimodal factorizations and the type-\(B\) Coxeter-Knuth relation of reduced words. We also give explicit algorithms for odd Kashiwara operators on signed unimodal factorizations of reduced words of type \(B\).