Crystal basis theory for the queer Lie superalgebra was developed in [9,10], where it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply qcrystal structure. In this paper, we explore the q-crystal structure of primed tableaux [13] (semistandard marked shifted tableaux [4]) and that of signed unimodal factorizations of reduced words of type B [13]. We give the 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 the explicit algorithms for odd Kashiwara operators on signed unimodal factorizations of reduced words of type B.
We show that the set of increasing factorizations of fixed-point-free (FPF) involution words has the structure of queer supercrystals. By exploiting the algorithm of symplectic shifted Hecke insertion recently introduced by Marberg (http://arxiv.org/abs/1901.06771v3), we establish the one-to-one correspondence between the set of increasing factorizations of fixed-point-free involution words and the set of primed tableau (semistandard marked shifted tableaux) and the latter admits the structure of queer supercrystals. In order to establish the correspondence, we prove that the Coxeter–Knuth related FPF-involution words have the same insertion tableau in the symplectic shifted Hecke insertion, where the insertion tableau is an increasing shifted tableau and the recording tableau is a primed tableau.
The branching coefficients of the tensor product of finite-dimensional irreducible Uq(g)-modules, where g is so(2n + 1, C) (Bn-type), sp(2n, C) (Cntype), and so(2n, C) (Dn-type), are expressed in terms of Littlewood-Richardson (LR) coefficients in the stable region. We give an interpretation of this relation by Kashiwara's crystal theory by providing an explicit surjection from the LR crystal of type Cn to the disjoint union of Cartesian product of LR crystals of A n−1 -type and by proving that LR crystals of types Bn and Dn are identical to the corresponding LR crystal of type Cn in the stable region.
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