Factoriality and class groups of cluster algebras

Abstract

Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-1 prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss-Leclerc-Schröer.

A geometric $q$-character formula for snake modules

“…In this section, we apply the results of [11] to show that the C 1 cluster algebra is factorial for Dynkin types A, D, E.…”

“…Then, for each i, we have x i x ′ i = f i , where f i is a binomial in the initial seed. Recall from [11] that two vertices i, j ∈ {1, 2, . .…”

“…Proof. By Corollary 5.2 of [11], we only need to show that every partner set in Q is a singleton. This holds because for every i ∈ I the exchange polynomial f i is a polynomial in the variable y i .…”

“…Lastly, the study of square free denominators led us to questions of factorization in the cluster algebra. Applying the results of [11], we include a proof that the C 1 cluster algebras of Hernandez-Leclerc are factorial for Dynkin types A, D, E.…”

“…They also showed that the cluster variables in a factorial cluster algebra are prime elements. In [11,Theorem 3.10], it was shown that if A is a factorial cluster algebra and x is a non-initial cluster variable, then the associated F -polynomial F x is prime.…”