We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors.

我们为契诃夫-夏皮罗广义集群代数构建了散射图，其中交换多项式被分解为二项式，推广了 Gross、Hacking、Keel 和 Kontsevich 的集群散射图。它们原来是出现在福克和冈察洛夫的集群二象性中的自然物体。在普通情况下，类似的特征和结构（例如阳性和集群复结构）也出现在广义情境中。在这些散射图的帮助下，我们表明广义集群变量是 θ 函数，因此相对于二项式因子中的系数具有一定的正性。