Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Fan, Qin

doi:10.1215/00127094-2017-0006

Cited by 76 publications

(145 citation statements)

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“…We remark that the partial order on monomials (2.2) therefore matches the partial order on Laurent monomials in initial cluster variables, as defined for arbitrary cluster algebras whose extended exchange matrices have full rank by Qin [56] (see also [8]). We believe that (2.2) was in fact an inspiration for Qin's definition of this partial order.…”

Section: 5mentioning

confidence: 88%

“…Making use of a well-known grading on C[Gr(n, m)], for each tableau T we define a homogeneous lift of ch(T ) from C[Gr(n, m, ∼)] to a localization of C[Gr(n, m)] (a priori, the lifts might have frozen variables in the denominator, so they naturally live in a localization). By deep results of Kashiwara, Kim, Oh, and Park [37] and Qin [56], we have the following. Theorem 1.2 (Theorem 3.25).…”

mentioning

confidence: 93%

“…We use brackets [S] to denote the Grothendieck class of an object S ∈ C ℓ . Qin [56] proved that for g of type A, D, E, every cluster monomial (resp. cluster variable) in K 0 (C g ℓ ) is a simple (resp.…”

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confidence: 99%

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Quantum affine algebras and Grassmannians

et al. 2020

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We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory C ℓ of U q ( sl n )-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [n + ℓ + 1]. Via the isomorphism, we define an element ch(T ) in a Grassmannian cluster algebra for every rectangular tableau T . By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T ) for some T . Using a formula of Arakawa-Suzuki, we give an explicit expression for ch(T ), and also give explicit q-character formulas for finite-dimensional U q ( sl n )-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.

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Section: 5mentioning

confidence: 88%

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confidence: 93%

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Quantum affine algebras and Grassmannians

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“…We emphasize that the maps g L S and φ L µ k (S ),S , which seem new, are as important as g R S and φ R µ k (S ),S . These properties are strongly related with the notion of common triangular bases introduced by Fan Qin [19]. For example, the order ≺ S is the same as the dominance order given in [19] and the bijectivity of g R S is one of the four conditions presented in [19] for the upper global basis to be a common triangular basis.…”

Section: Introductionmentioning

confidence: 95%

Laurent phenomenon and simple modules of quiver Hecke algebras

Kashiwara¹

2019

Compositio Math.

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In this paper, we study consequences of the results of Kang et al. ([13]) on a monoidal categorification of the unipotent quantum coordinate ring A q (n(w)) together with the Laurent phenomenon of cluster algebras. We show that if a simple module S in the category C w strongly commutes with all the cluster variables in a cluster [C ], then [S] is a cluster monomial in [C ]. If S strongly commutes with cluster variables except exactly one cluster variable [M k ], then [S] is either a cluster monomial in [C ] or a cluster monomial in µ k ([C ]). We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Fan Qin ([19])) of the localization A q (n(w)) of A q (n(w)) at the frozen variables. A characterization on the commutativity of a simple module S with cluster variables in a cluster [C ] is given in terms of the denominator vector of [S] with respect to the cluster [C ].

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“…There are monoidal subcategories C N (N ∈ Z 1 ), C − g and C 0 g of C g , introduced by Hernandez-Leclerc in [22,24] (see also [37] for C 0 g ), whose Grothendieck rings K(C) have cluster algebra structures, and which are conjectured to be monoidal categorifications of K(C). The conjecture for C 0 g of affine types A (t) n (t = 1, 2) and B (1) n is proved in [37] indirectly by using generalized quantum Schur-Weyl duality constructed in [26,29,35], and for C 1 and C N (N ∈ Z 1 ) of untwisted affine types ADE are proved in [22,23,47] and [50] respectively, by approaches different from [37]. However, by the lack of Z-grading structure on U ′ q (g), one can not apply the framework in [30] to those categories for monoidal categorifications directly (see also [4,Section 4]).…”

Section: Introductionmentioning

confidence: 99%