Crystal bases for the quantum queer superalgebra

Grantcharov, Dimitar; Jung, Ji Hye; Kang, Seok Jin; Kashiwara, Masaki

doi:10.4171/jems/540

Cited by 28 publications

(44 citation statements)

References 22 publications

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“…Remark 2.3. These rules are different from the ones given in [5,9,10] because we adopt the anti-Kashiwara convention for the tensor product rule.…”

Section: Crystals For the Queer Lie Superalgebramentioning

confidence: 99%

“…Several combinatorial models have been proposed and investigated in the study of Schur P -or Q-functions [4,5,9,10,21,22,23]. Among them, semistandard decomposition tableaux were shown to admit the structure of crystals for the queer Lie superalgebra or simply q-crystal structure by Grantcharov et al [9,10]. On the other hand, Hawkes et al [13] gave a bijection between the set of semistandard unimodal tableaux and that of primed tableaux of the same shape.…”

Section: Introductionmentioning

confidence: 99%

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mentioning

confidence: 99%

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

Hiroshima

2019

Publ. Res. Inst. Math. Sci.

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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.

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“…Remark 2.3. These rules are different from the ones given in [5,9,10] because we adopt the anti-Kashiwara convention for the tensor product rule.…”

Section: Crystals For the Queer Lie Superalgebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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

Hiroshima

2019

Publ. Res. Inst. Math. Sci.

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“…The challenge of working over the field C(q) is that the classical limit of an irreducible highest Uq(qn)-supermodule may no longer be irreducible. As indicated in [7], enlarging the base field to C((q)) will overcome this challenge.…”

Section: Introductionmentioning

confidence: 99%

Howe duality for quantum queer superalgebras

Chang

Wang

2020

Journal of Algebra

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We establish a new Howe duality between a pair of quantum queer superalgebras (U q −1 (qn), Uq(qm)). The key ingredient is the construction of a non-commutative analogue Aq(qn, qm) of the symmetric superalgebra S(C mn|mn ) with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a U q −1 (qn) ⊗ Uq(qm)supermodule structure that admits a multiplicity-free decomposition. We also show that the (U q −1 (qn), Uq(qm))-Howe duality implies the Sergeev-Olshanski duality.MSC(2010): 17B37, 17D10, 20G42.

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“…In type B, there are two related classes of tableaux, 'P-tableaux' and 'Q-tableaux', enumerated respectively by the Schur Pand Q-functions. A crystal structure on P-tableaux, corresponding to the representation theory of the quantum queer superalgebra q(n), was introduced in [6], and the combinatorics and structure of these crystals were further studied in [1], [7], and [10]. In [5], Choi and Kwon use the structure to understand Schur P-positivity of certain skew Schur functions.The crystals studied in this paper are on Q-tableaux and are nonisomorphic to the q(n) crystals.…”

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confidence: 99%

Axioms for Shifted Tableau Crystals

Gillespie¹,

Levinson²

2019

Electron. J. Combin.

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We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed in [8]. These axioms closely resemble those developed by Stembridge [18] for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur Q-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.This result gives a method to show that a symmetric function is Schur-positive: one introduces operators e i , f i on the underlying set, satisfying the local axioms. Theorem 1.1 then implies that the weight generating function is Schur-positive, and enumerating the highest weight elements of the connected components gives the coefficients in the Schur expansion. This method has recently been applied successfully by Morse and Schilling [13] to certain affine Stanley symmetric functions.In this paper, we give analogous local axioms for the crystal-like structure introduced in [8] on shifted semistandard tableaux. We write = ShST(λ/µ, n) for the set of shifted semistandard tableaux of shifted skew shape λ/µ and entries ≤ n. The raising and lowering operators defined in [8] were introduced to answer geometric questions involving the cohomology of the odd orthogonal Grassmannian H * (OG(n, V )), where V is a (2n + 1)-dimensional complex vector space with a nondegenerate symmetric form [9]. Combinatorially, these operators are coplactic, that is, their action is invariant under shifted jeu de taquin slides, and this property essentially determines their action on all shifted skew semistandard tableaux (see Sections 2 and 5 for additional discussion).Interestingly, although the crystals' combinatorial and enumerative properties describe type B Schubert calculus, the crystal itself resembles a 'doubled' type A crystal, with two sets of operators e i , f i and e i , f i . Accordingly, our axioms resemble a 'doubled' form of the Stembridge axioms.Remark 1.2. In type B, there are two related classes of tableaux, 'P-tableaux' and 'Q-tableaux', enumerated respectively by the Schur Pand Q-functions. A crystal structure on P-tableaux, corresponding to the representation theory of the quantum queer superalgebra q(n), was introduced in [6], and the combinatorics and structure of these crystals were further studied in [1], [7], and [10]. In [5], Choi and Kwon use the structure to understand Schur P-positivity of certain skew Schur functions.The crystals studied in this paper are on Q-tableaux and are nonisomorphic to the q(n) crystals. The Schur Pand Q-functions, first defined by Schur [16], are dual classes of symmetric functions under the standard Hall inner product [12], and it would be interesting to understand whether these crystals are also in some sense 'dual' to each other.Our main theorem is as follows. Theorem 1.3. Let G be a directed graph with vertices weighted by n ≥0 and edges labeled by 1 , 1, . . . , (n−1) , n−1. Suppose G satisfies the axioms stated in section 3 below.Then every connec...

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Crystal bases for the quantum queer superalgebra

Cited by 28 publications

References 22 publications

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

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

Howe duality for quantum queer superalgebras

Axioms for Shifted Tableau Crystals

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