$\mathfrak{q}$-Crystal Structure on Primed Tableaux and on Signed Unimodal Factorizations of Reduced Words of Type $B$

Abstract

Crystal basis theory for the queer Lie superalgebra was developed in Grantcharov et al. (Trans. Amer. Math. Soc. 366, 2013, 457–489) and Grantcharov et al. (J. Europ. Math. Soc. 17, 2015, 1593–1627), 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 (Hawkes et al., Electron. J. Combin. 24, 2017, P3.51) (semistandard marked shifted tableaux; Cho, Trans. Amer. Math. Soc. 365, 2013, 939–972) and that of signed unimodal factorizations of reduced words of type $B$ (Hawkes et al., Electron. J. Combin. 24 (2017) P3.51). 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 $B$.