Scattering diagrams for generalized cluster algebras

IF 0.9 1区数学 Q2 MATHEMATICS

Algebra & Number Theory Pub Date : 2024-10-21 DOI:10.2140/ant.2024.18.2179

Lang Mou

引用次数: 0

Abstract

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.

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广义簇代数的散射图

我们构建了契诃夫-夏皮罗广义簇代数的散点图，其中交换多项式被因子化为二项式，从而推广了格罗斯、哈金、基尔和康采维奇的簇散点图。它们是福克和冈察洛夫的簇对偶中出现的自然对象。普通情况下的类似特征和结构（如正性和簇复合结构）也出现在广义情况下。借助这些散点图，我们证明了广义簇变量是 Theta 函数，因此相对于二项式因子中的系数具有一定的正性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Algebra & Number Theory MATHEMATICS-

CiteScore

1.80

自引率

7.70%

发文量

审稿时长

6-12 weeks

期刊介绍： ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.