The valuation pairing on an upper cluster algebra

• Published in: Journal für die reine und angewandte Mathematik Vol. 2024; no. 806; pp. 71 - 114
• Main Authors: Cao, Peigen, Keller, Bernhard, Qin, Fan
• Format: Journal Article
• Language: English
• Published: De Gruyter 01.01.2024
• ISSN: 0075-4102; 1435-5345
• DOI: 10.1515/crelle-2023-0080

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 : any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in and another element not divisible by the cluster variables in . 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, -vectors, -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 -vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its -polynomial; the -polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.