The valuation pairing on an upper cluster algebra

Abstract: It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed t: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in t and another element not divisible by the cluster variables in t. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, d-vectors, F-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of d-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its F-polynomial; the F-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• Extent: Online-Ressource
• Language: Englisch

References and Relationships

• Bibliographic citation: The valuation pairing on an upper cluster algebra ; volume:2024 ; number:806 ; year:2024 ; pages:71-114 ; extent:44 ; Journal für die reine und angewandte Mathematik ; 2024, Heft 806 (2024), 71-114 (gesamt 44)

Contributors, Places and Time

• Creator: Cao, Peigen ; Keller, Bernhard ; Qin, Fan

Further information

• DOI: 10.1515/crelle-2023-0080
• URN: urn:nbn:de:101:1-2024010813024940501274
• Rights: Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Last update: 15.08.2025, 7:33 AM CEST