The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators". After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to develop an algorithm which explicitly calculate these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section.

中文翻译：

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椭圆曲线的霍尔代数和 Hecke 算子图

Hecke 算子的图形编码了关于这个算子在全局函数域上的自守形式上的动作的所有信息。这些图是 Lorscheid 在他的 $\text{PGL}_{2}$ 博士论文中引入的，我们在论文“On graphs of Hecke operators”中推广到 $\text{GL}_{n}$。在回顾了一些一般性质之后，我们解释了函数场与霍尔代数的联系。在椭圆函数场的情况下，我们可以使用 Burban-Schiffmann 和 Fratila 的结构结果来开发一种算法来明确计算这些图。我们应用这个算法来确定一些结构常数，并在最后一节中明确提供了排名二的情况。