1. The valuation pairing on an upper cluster algebra.
- Author
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Cao, Peigen, Keller, Bernhard, and Qin, Fan
- Subjects
- *
CLUSTER algebras , *VALUATION , *COMBINATORICS , *FACTORIZATION , *MULTIPLICITY (Mathematics) - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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