$\mathfrak{q}$-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 [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.

Highest weight crystals for Schur Q-functions

“…This section describes a q + n -crystal on factorizations of certain analogues of reduced words for involutions in symmetric groups. This structure extends a q n -crystal studied in [19,31], which is itself based on a gl n -crystal described in [34].…”

“…For the precise definitions of q n -crystals and their tensor product, see Section 3.2. Besides in [11,12,13], these crystals have been studied in [2,6,10,19,20,31], for example. Normal q n -crystals are defined in terms of tensor powers of the standard q n -crystal in the same way as in the gl n -case.…”

“…Next, we show in Section 6 how to extend a q n -crystal structure on semistandard shifted tableaux studied in [2,17,19] to a q + n -crystal on a larger set. Building on results in [19], we are able to prove that each q + n -crystal of shifted tableaux of a fixed strict partition shape is connected with unique highest and lowest weight elements; see Theorem 6. 20.…”

A crystal-like structure on shifted tableaux

“…In contrast to the elegance of crystal operators on ordinary semistandard tableaux, similar structures using shifted tableaux have proven more complex. In representation theory, prior work first used semistandard decomposition tableaux [10,21], then shifted tableaux [2,5,6,12] to understand the quantum queer superalgebra q(n), though the corresponding crystal operators are not direct analogues of the coplactic operators in type A. Shifted tableaux also arise in the context of projective representations of the symmetric group [23], and in Schubert calculus in types B and C [16], pertaining to the Schur P polynomials [4,22].…”

Characterization of queer supercrystals

“…Choi and Kwon [CK18] provide a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur P -functions by using the theory of q(n)-crystals. Independently, Hiroshima [Hir18] and Assaf and Oguz [AKO18a, AKO18b] defined a queer crystal structure on semistandard shifted tableaux, extending the type A crystal structure of [HPS17] on these tableaux.…”