发表机构
University of California, Los Angeles（加州大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明热带T-系统中沿二部带的红蓝突变统计量具有同均值性，红突变数等于根系中根数，并引入Zamolodchikov扇，解决第一作者的猜想。
AI 中文摘要
2007年，Fomin和Zelevinsky引入了二部带（bipartite belt），即一系列二部簇突变，其交换关系构成一个离散动力系统。对于每个Dynkin图，该系统是周期的，这是Zamolodchikov周期性的一个特例。在相关的热带动力学中，沿带的每次突变可根据热带交换关系中哪一项达到最大值而自然地染成红色或蓝色。这给出了热带轨道上的红蓝统计量。我们证明该统计量是同均值的（homomesic），即红突变和蓝突变的平均数与轨道无关。此外，红突变的数目由相关根系中根的数目给出。这解决了第一作者的一个猜想。在此过程中，我们引入了Zamolodchikov扇，它与正热带Grassmannian密切相关，并且可能具有独立的研究意义。
英文摘要
In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite cluster mutations whose exchange relations form a discrete dynamical system. For each Dynkin diagram, this system is periodic, as a special case of Zamolodchikov periodicity. In the associated tropical dynamics, every mutation along the belt can be naturally colored red or blue, according to which term attains the maximum in the tropical exchange relation. This gives a red-blue statistic on tropical orbits. We prove that this statistic is homomesic, i.e. that the average numbers of red and blue mutations are independent of the orbit. Moreover, the number of red mutations is given by the number of roots in the associated root system. This resolves a conjecture of the first author. In the process we introduce the Zamolodchikov fan, which is closely related to the positive tropical Grassmannian, and may be of independent interest.
Comments26 pages, 14 figures
