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Mutation commutativity between geometric realizations and related quivers of classical cluster algebras and applications.

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Τίτλος: Mutation commutativity between geometric realizations and related quivers of classical cluster algebras and applications.
Συγγραφείς: Zhang, Baojin1 (AUTHOR), Tang, Xiaomin1 (AUTHOR) x.m.tang@163.com
Πηγή: Journal of Algebra & Its Applications. Mar2026, Vol. 25 Issue 3, p1-33. 33p.
Θεματικοί όροι: *Cluster algebras, *Mutations (Algebra), *Geometric modeling, *Root systems (Algebra), *Triangulation, *Graph theory
Περίληψη: Cluster algebras, introduced by Fomin and Zelevinsky, are connected to dual canonical bases. Geometric realization is an effective way to solve algebraic problems. In 2002, Fomin and Zelevinsky proposed geometric realizations for root systems and finite-type cluster algebras. This study examines the regulation of mutation in geometric realization. We establish a fascinating link between centrally symmetric triangulations and quivers in classical cluster algebras, showing that these triangulations and quivers are interchangeable under mutation. This connection leads to a significant finding: any cluster variable of a given cluster algebra can induce a connected subquiver within the corresponding initial quiver. Furthermore, we demonstrate that any two quivers generated by cluster variables intersect, emphasizing their connectedness. To illustrate the practical implications of this geometric realization, we present a convenient method to compute seed mutations of type D n and define a category based on centrally symmetric triangulations. This category's objects are ℤ -linear combinations of positive roots of the root system of type D n , and we propose an equivalent condition for morphisms. This condition is a generalization of the case of type A n by [P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc.358(3) (2006) 1347–1364]. Finally, we propose a geometric realization for cluster algebras: weighted triangulations. This approach does not require cluster algebras to be of finite type and can verify whether a matrix from a cluster algebra is equivalent to a subdiagonal matrix. [ABSTRACT FROM AUTHOR]
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  Data: Mutation commutativity between geometric realizations and related quivers of classical cluster algebras and applications.
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  Data: <searchLink fieldCode="AR" term="%22Zhang%2C+Baojin%22">Zhang, Baojin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Tang%2C+Xiaomin%22">Tang, Xiaomin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> x.m.tang@163.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Algebra+%26+Its+Applications%22">Journal of Algebra & Its Applications</searchLink>. Mar2026, Vol. 25 Issue 3, p1-33. 33p.
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  Data: *<searchLink fieldCode="DE" term="%22Cluster+algebras%22">Cluster algebras</searchLink><br />*<searchLink fieldCode="DE" term="%22Mutations+%28Algebra%29%22">Mutations (Algebra)</searchLink><br />*<searchLink fieldCode="DE" term="%22Geometric+modeling%22">Geometric modeling</searchLink><br />*<searchLink fieldCode="DE" term="%22Root+systems+%28Algebra%29%22">Root systems (Algebra)</searchLink><br />*<searchLink fieldCode="DE" term="%22Triangulation%22">Triangulation</searchLink><br />*<searchLink fieldCode="DE" term="%22Graph+theory%22">Graph theory</searchLink>
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  Data: Cluster algebras, introduced by Fomin and Zelevinsky, are connected to dual canonical bases. Geometric realization is an effective way to solve algebraic problems. In 2002, Fomin and Zelevinsky proposed geometric realizations for root systems and finite-type cluster algebras. This study examines the regulation of mutation in geometric realization. We establish a fascinating link between centrally symmetric triangulations and quivers in classical cluster algebras, showing that these triangulations and quivers are interchangeable under mutation. This connection leads to a significant finding: any cluster variable of a given cluster algebra can induce a connected subquiver within the corresponding initial quiver. Furthermore, we demonstrate that any two quivers generated by cluster variables intersect, emphasizing their connectedness. To illustrate the practical implications of this geometric realization, we present a convenient method to compute seed mutations of type D n and define a category based on centrally symmetric triangulations. This category's objects are ℤ -linear combinations of positive roots of the root system of type D n , and we propose an equivalent condition for morphisms. This condition is a generalization of the case of type A n by [P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc.358(3) (2006) 1347–1364]. Finally, we propose a geometric realization for cluster algebras: weighted triangulations. This approach does not require cluster algebras to be of finite type and can verify whether a matrix from a cluster algebra is equivalent to a subdiagonal matrix. [ABSTRACT FROM AUTHOR]
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1142/S0219498825503530
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 33
        StartPage: 1
    Subjects:
      – SubjectFull: Cluster algebras
        Type: general
      – SubjectFull: Mutations (Algebra)
        Type: general
      – SubjectFull: Geometric modeling
        Type: general
      – SubjectFull: Root systems (Algebra)
        Type: general
      – SubjectFull: Triangulation
        Type: general
      – SubjectFull: Graph theory
        Type: general
    Titles:
      – TitleFull: Mutation commutativity between geometric realizations and related quivers of classical cluster algebras and applications.
        Type: main
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          Name:
            NameFull: Zhang, Baojin
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          Name:
            NameFull: Tang, Xiaomin
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          Dates:
            – D: 01
              M: 03
              Text: Mar2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 02194988
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              Value: 25
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Journal of Algebra & Its Applications
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