Atomic bases of cluster algebras of types \(A\) and \(\tilde A\) (Q2856117)

The notion of a cluster algebra was invented by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups. Let \(Q\) be a finite connected quiver with \(n\) vertices and without oriented cycles of length \(1\) and \(2\) and \({{x}} = (x_1, \ldots, x_n)\) be a \(n\)-tuple of variables. The pair \((Q,{{x}})\) is called the {cluster} of the seed. Through {mutation}, one can define recursively a family of seeds. The (coefficient--free) {cluster algebra }\(\mathcal A_Q\) is the \(\mathbb{Z}\)-subalgebra of the {ambient field} \(\mathbb{Q}(x_1, \ldots, x_n)\) generated by all the clusters of the seeds arising from mutation. The so--called Laurent phenomenon tells that \(\mathcal A_Q\) is a subring of \(\mathbb{Z}[c_1^{\pm 1}, \ldots, c_n^{\pm 1}]\) for any cluster \({{c}}= (c_1, \ldots, c_n)\) in \(\mathcal{A}_Q\). An element in \(\mathcal{A}_Q\) is called {positive} if it belongs to the semiring \(\mathbb{Z}_{\geq 0}[c_1^{\pm 1}, \ldots, c_n^{\pm 1}]\) for any cluster \({{c}} = (c_1, \ldots, c_n)\) in \(\mathcal{A}_Q\). The cone of positive elements in \(\mathcal A_Q\) is denoted by \(\mathcal A_Q^+\). An {atomic basis} (or a {canonically positive basis}) of \(\mathcal{A}_Q\) is a \(\mathbb{Z}\)-basis \(\mathcal{B}\) of \(\mathcal{A}_Q\) such that \(\mathcal{A}_Q^+ = \bigoplus_{b \in \mathcal{B}} \mathbb{Z}_{\geq 0}b.\) The problem of showing the existence of the atomic basis of \(\mathcal{A}_Q\) remains wide open in general.NEWLINENEWLINERecently, Cerulli proved that the atomic basis coincides with the set of cluster monomials of \(\mathcal{A}_Q\) if \(\mathcal{A}_Q\) is of finite type [\textit{G. Cerulli Irelli}, ``Positivity in skew-symmetric cluster algebras of finite type'', \url{arXiv:1102.3050}]. If \(\mathcal A_Q\) is not of finite type, the statement is not true. For the particular cases when \(Q\) is an affine quiver of type \(\widetilde A_{1,1}\) or \(\widetilde A_{2,1}\), the atomic bases were explicitly constructed in [\textit{G. Cerulli Irelli}, Algebr. Represent. Theory 15, No. 5, 977--1021 (2012; Zbl 1261.13015)] and [\textit{P. Sherman} and \textit{A. Zelevinsky}, Mosc. Math. J. 4, No. 4, 947--974 (2004; Zbl 1103.16018)], respectively. The main result of the paper is the generalization of this construction to arbitrary quivers of affine type \(\widetilde A\). Meanwhile, the authors also provided a new, short and elementary proof of Cerulli's result for cluster algebras of type \(A\).