Cluster algebras and snake modules

Duan, Bing; Lǐ, Jiànróng; Luo, Yanfeng

doi:10.1016/j.jalgebra.2018.10.027

Cited by 12 publications

(22 citation statements)

References 25 publications

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“…This proves Conjecture 1.2 for snake modules and gives a geometric algorithm for the truncated q-characters. As a slight generalization of Theorem 3.4 of [10], we show in Theorem 4.2 that snake modules are real modules.…”

Section: Snake Modulesmentioning

confidence: 79%

“…In [24], they introduced a purely combinatorial method to compute q-characters for snake modules of types A and B, and in [25], they used snake modules to construct extended T -systems for types A and B. In [10], it was shown that all prime snake modules are real and that they correspond to some cluster variables in the cluster algebra constructed by Hernandez and Leclerc. Our first main theorem is the following. This proves Conjecture 1.2 for snake modules and gives a geometric algorithm for the truncated q-characters.…”

Section: Snake Modulesmentioning

confidence: 99%

“…A graphic interpretation is that in

G^{-}

, the upper bound of the integer

j_{ℓ}

is the minimum distance from the columns with vertex labelings

(i_{ℓ}, -)

and

(i_{ℓ + 1}, -)

to the leftmost column and the rightmost column, respectively. The value of

j_{ℓ}

is associated to the multiplicity of the left and right operators defined in [10, Section 5.5].…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, in [25, Theorem 4.1], Mukhin and Young introduced a set of 3‐term recurrence relations satisfied by

q

‐characters of prime snake modules, called extended

T

‐system, which generalizes the usual

T

‐system. Moreover, in [10, Theorem 4.1], the authors introduced a system of equations satisfied by

q

‐characters of prime snake modules, called

S

‐system, which contains the usual

T

‐system. In fact, the equations in the

S

‐system can be interpreted as cluster transformations in the cluster algebra

scriptA

where the initial cluster variables correspond to certain Kirillov–Reshetikhin modules.…”

Section: Preliminariesmentioning

confidence: 99%

“…In [24], they introduced a purely combinatorial method to compute

q

‐characters for snake modules of types

double-struckA

and

double-struckB

, and in [25], they used snake modules to construct extended

T

‐systems for types

double-struckA

and

double-struckB

. In [10], it was shown that all prime snake modules are real and that they correspond to some cluster variables in the cluster algebra constructed by Hernandez and Leclerc.…”

Section: Introductionmentioning

confidence: 99%

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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: 79%

Section: Snake Modulesmentioning

confidence: 99%

“…A graphic interpretation is that in

G^{-}

, the upper bound of the integer

j_{ℓ}

is the minimum distance from the columns with vertex labelings

(i_{ℓ}, -)

and

(i_{ℓ + 1}, -)

to the leftmost column and the rightmost column, respectively. The value of

j_{ℓ}

is associated to the multiplicity of the left and right operators defined in [10, Section 5.5].…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, in [25, Theorem 4.1], Mukhin and Young introduced a set of 3‐term recurrence relations satisfied by

q

‐characters of prime snake modules, called extended

T

‐system, which generalizes the usual

T

‐system. Moreover, in [10, Theorem 4.1], the authors introduced a system of equations satisfied by

q

‐characters of prime snake modules, called

S

‐system, which contains the usual

T

‐system. In fact, the equations in the

S

‐system can be interpreted as cluster transformations in the cluster algebra

scriptA

where the initial cluster variables correspond to certain Kirillov–Reshetikhin modules.…”

Section: Preliminariesmentioning

confidence: 99%

“…In [24], they introduced a purely combinatorial method to compute

q

‐characters for snake modules of types

double-struckA

and

double-struckB

, and in [25], they used snake modules to construct extended

T

‐systems for types

double-struckA

and

double-struckB

. In [10], it was shown that all prime snake modules are real and that they correspond to some cluster variables in the cluster algebra constructed by Hernandez and Leclerc.…”

Section: Introductionmentioning

confidence: 99%

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A geometric q‐character formula for snake modules

Duan

Schiffler

2020

J. London Math. Soc.

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Strong Duality Data of Type A and Extended T-Systems

Naoi

2024

Transformation Groups

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The extended T-systems are a number of relations in the Grothendieck ring of the category of finite-dimensional modules over the quantum affine algebras of types $$A_n^{(1)}$$ A n ( 1 ) and $$B_n^{(1)}$$ B n ( 1 ) , introduced by Mukhin and Young as a generalization of the T-systems. In this paper we establish the extended T-systems for more general modules, which are constructed from an arbitrary strong duality datum of type A. Our approach does not use the theory of q-characters, and so also provides a new proof to the original Mukhin–Young’s extended T-systems.

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