Factoriality and class groups of cluster algebras

Elsener, Ana Garcia; Lampe, Philipp; Smertnig, Daniel

doi:10.1016/j.aim.2019.106858

Cited by 10 publications

(16 citation statements)

References 21 publications

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“…In this section, we apply the results of [16] to show that the C 1 cluster algebra is factorial for Dynkin types A, D, E.…”

Section: Factorial Cluster Algebrasmentioning

confidence: 99%

“…Then, for each i, we have x i x i = f i , where f i is a binomial in the initial seed. Recall from [16] that two vertices i, j ∈ {1, 2, . .…”

Section: Factorial Cluster Algebrasmentioning

confidence: 99%

“…Then, for each

i

, we have

x_{i} x_{i}^{'} = f_{i}

, where

f_{i}

is a binomial in the initial seed. Recall from [16] that two vertices

i, j \in {1, 2, \dots, n}

are called partners if

f_{i}

and

f_{j}

have a non‐trivial common factor. Partnership is an equivalence relation and the equivalence classes are called partner sets .…”

Section: Factorial Cluster Algebrasmentioning

confidence: 99%

“…Finally, the study of square‐free denominators led us to questions of factorization in the cluster algebra. Applying the results of [16], we include a proof that the

C_{1}

cluster algebras of Hernandez–Leclerc are factorial for Dynkin types

A, D, E

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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“…In this section, we apply the results of [16] to show that the C 1 cluster algebra is factorial for Dynkin types A, D, E.…”

Section: Factorial Cluster Algebrasmentioning

confidence: 99%

“…Then, for each i, we have x i x i = f i , where f i is a binomial in the initial seed. Recall from [16] that two vertices i, j ∈ {1, 2, . .…”

Section: Factorial Cluster Algebrasmentioning

confidence: 99%

“…Then, for each

i

, we have

x_{i} x_{i}^{'} = f_{i}

, where

f_{i}

is a binomial in the initial seed. Recall from [16] that two vertices

i, j \in {1, 2, \dots, n}

are called partners if

f_{i}

and

f_{j}

have a non‐trivial common factor. Partnership is an equivalence relation and the equivalence classes are called partner sets .…”

Section: Factorial Cluster Algebrasmentioning

confidence: 99%

“…Finally, the study of square‐free denominators led us to questions of factorization in the cluster algebra. Applying the results of [16], we include a proof that the

C_{1}

cluster algebras of Hernandez–Leclerc are factorial for Dynkin types

A, D, E

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A geometric q‐character formula for snake modules

Duan

Schiffler

2020

J. London Math. Soc.

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show abstract

“…Every transfer homomorphism is a weak transfer homomorphism and the converse holds if H and B are both commutative ([5, Section 2]; in general, this is not true as can be seen from [3]). If θ : H → B is a weak transfer homomorphism, then it is easy to check (e.g., [ [25].…”

Section: Transfer Krull Monoidsmentioning

confidence: 99%

Sets of arithmetical invariants in transfer Krull monoids

Geroldinger

Zhong

2019

Journal of Pure and Applied Algebra

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Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between 1 and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.

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A realization result for systems of sets of lengths

Geroldinger

Zhong

2021

Isr. J. Math.

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Factoriality and class groups of cluster algebras

Cited by 10 publications

References 21 publications

A geometric q‐character formula for snake modules

A geometric q‐character formula for snake modules

Sets of arithmetical invariants in transfer Krull monoids

A realization result for systems of sets of lengths

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