Title: T-systems, networks and dimers Show Chinese title

Title: T系统，网络和镶嵌

Abstract: We study the solutions of the T-system for type A, also known as the octahedron equation, viewed as a 2+1-dimensional discrete evolution equation. These may be expressed entirely in terms of the stepped surface over which the initial data are specified, via a suitably defined flat $GL_n$ connection which embodies the integrability of this infinite rank system. By interpreting the connection as the transfer operator for a directed graph or network with weighted edges, we show that the solution at a given point is expressed as the partition function for dimers on a bipartite graph dual to the "shadow" of the point onto the initial data stepped surface. We extend the result to the case of other geometries such as that of the evaporation of a cube corner crystal, and to a reformulation of the Kenyon-Pemantle discrete hexahedron equation. Show Chinese abstract

Abstract: 我们研究类型A的T系统，也称为八面体方程，将其视为2+1维离散演化方程。 这些解可以完全通过指定初始数据的阶梯表面来表达，通过一个适当定义的平坦$GL_n$连接，该连接体现了这个无限秩系统的可积性。 通过将连接解释为具有加权边的有向图或网络的转移算子，我们证明了在给定点的解可以表示为在与该点到初始数据阶梯表面的“阴影”对偶的二分图上的偶配的划分函数。 我们将结果扩展到其他几何情况，例如立方体角晶体的蒸发情况，以及Kenyon-Pemantle离散六面体方程的一种重新表述。