M-Systems and Cluster Algebras

Zhang, Qianqian; Duan, Bing; Lǐ, Jiànróng; Luo, Yanfeng

doi:10.1093/imrn/rnv287

Cited by 10 publications

(10 citation statements)

References 44 publications

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“…where δ ij is the Kronecker delta, and we use the convention 0 =1 n = 0. In particular, L(m) is a prime snake module if j satisfies the following bounded conditions: (2) Minimal affinizations introduced in [3] (see also [32]) are snake modules, by taking…”

Section: Snake Modulesmentioning

confidence: 99%

A geometric q‐character formula for snake modules

Duan

Schiffler

2020

J. London Math. Soc.

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Let C be the category of finite-dimensional modules over the quantum affine algebra Uq(g) of a simple complex Lie algebra g. Let C − be the subcategory introduced by Hernandez and Leclerc. We prove the geometric q-character formula conjectured by Hernandez and Leclerc in types A and B for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the F-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square-free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category C1 are factorial for Dynkin types A, D, E.

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Section: Snake Modulesmentioning

confidence: 99%

A geometric q‐character formula for snake modules

Duan

Schiffler

2020

J. London Math. Soc.

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“…These mutation sequences are defined in the same cluster algebra. In [34], the mutation sequences for minimal affinizations which satisfy i 1 < · · · < i r are defined in a cluster algebra A and the mutation sequences for minimal affinizations which satisfy i 1 > · · · > i r are defined in another cluster algebra A which is dual to A .…”

Section: 7mentioning

confidence: 99%

“…When M i is the highest weight monomial of a minimal affinization, the mutation sequence Seq i is similar to the mutation sequence for a minimal affinization in [34]. A minimal affinization is a U q g-module with a highest weight of the form (see Section 4.1):…”

Section: Introductionmentioning

confidence: 99%

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Cluster algebras and snake modules

2019

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Snake modules introduced by Mukhin and Young form a family of modules of quantum affine algebras. The aim of this paper is to prove that the Hernandez-Leclerc conjecture about monoidal categorifications of cluster algebras is true for prime snake modules of types A n and B n . We prove that prime snake modules are real. We introduce S-systems consisting of equations satisfied by the q-characters of prime snake modules of types A n and B n . Moreover, we show that every equation in the S-system of type A n (respectively, B n ) corresponds to a mutation in the cluster algebra A (respectively, A ′ ) constructed by Hernandez and Leclerc and every prime snake module of type A n (respectively, B n ) corresponds to some cluster variable in A (respectively, A ′ ). In particular, this proves that the Hernandez-Leclerc conjecture is true for all prime snake modules of types A n and B n .

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“…This is the second paper of a series based on a project aiming at describing the classification of the Drinfeld polynomials of the irregular minimal affinizations of type D. The theory of minimal affinizations, initiated in [1,10], is object of intensive study due to its rich structure and connections to other areas such as mathematical physics and combinatorics [5,13,17,18,19,22,28,29]. We refer to the first two paragraphs of the first paper of the series [26] for an account of the status of this classification problem when this project started.…”

Section: Introductionmentioning

confidence: 99%

Non-minimality of certain irregular coherent preminimal affinizations

Moura

Pereira

2018

Pacific J. Math.

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Let g be a finite-dimensional simple Lie algebra of type D or E and λ be a dominant integral weight whose support bounds the subdiagram of type D4. We study certain quantum affinizations of the simple g-module of highest weight λ which we term preminimal affinizations of order two (this is the maximal order for such λ). This class can be split in two: the coherent and the incoherent affinizations. If λ is regular, Chari and Pressley proved that the associated minimal affinizations belong to one of the three equivalent classes of coherent preminimal affinizations. In this paper we show that, if λ is irregular, the coherent preminimal affinizations are not minimal under certain hypotheses. Since these hypotheses are always satisfied if g is of type D4, this completes the classification of minimal affinizations for type D4 by giving a negative answer to a conjecture of Chari-Pressley stating that the coherent and the incoherent affinizations were equivalent in type D4 (this corrects the opposite claim made by the first author in a previous publication).

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M-Systems and Cluster Algebras

Cited by 10 publications

References 44 publications

A geometric q‐character formula for snake modules

A geometric q‐character formula for snake modules

Cluster algebras and snake modules

Non-minimality of certain irregular coherent preminimal affinizations

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