On class groups of upper cluster algebras
Mara Pompili
TL;DR
This paper develops a unified factorization-theoretic framework for (upper) cluster algebras by leveraging Krull-domain methods. It establishes that all cluster and upper cluster algebras are finite factorization domains (FF-domains), with cluster variables acting as strong atoms, and uses this to study the multiplicative structure via valuation pairings. For full rank upper cluster algebras, it computes the class group: under the starfish condition at a seed, $\mathcal{C}(\mathcal{U})$ is a free abelian group of rank $t-n$, where $t$ counts height-1 primes containing exchangeable variables; this yields a clear dichotomy between factorial (UFD) and non-factorial cases, linked to the irreducibility of exchange polynomials. Overall, the work connects exchange-polynomial data to divisor-class theory, providing concrete invariants and local-to-global factorization insights for upper cluster algebras.
Abstract
We compute the class group of a full rank upper cluster algebra in terms of its exchange polynomials. As a corollary, we recover a theorem by Cao, Keller, and Qin from 2023 characterizing the UFDs among these algebras. Furthermore, under the additional hypothesis of acyclicity, we obtain a result by Garcia Elsener, Lampe, and Smertnig from 2019. Moreover, we show that all cluster and upper cluster algebras are finite factorization domains, meaning that every non-unit factors as a product of atoms (or irreducibles) and, for each element, there are only finitely many such factorizations up to order and associates. This strengthens another result by Cao, Keller, and Qin showing that cluster and upper cluster algebras are atomic.