Positivity for Generalized Cluster Variables of Affine Quivers

It has been proved in Lee and Schiffler, Ann. of Math. 182(1) 73–125 2015 that cluster variables of all skew-symmetric cluster algebras are positive. i.e., every cluster variable as a Laurent polynomial in the cluster variables of any fixed cluster has positive coefficients. We prove that every regular generalized cluster variable of an affine quiver is positive. As a corollary, we obtain that generalized cluster variables of affine quivers are positive and we also construct various positive bases. This generalizes the results in Dupont, J. Algebra Appl. 11(4) 19 2012 and Ding et al. Algebr. Represent. Theory 16(2) 491–525 2013.