In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module, we define a Coxeter element in a symmetric group, and we establish a bijection between these lattices and the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondence between string modules and snake graphs, we give a new concise formulation of snake graph calculus.

中文翻译：

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字符串模块、蛇图和弱 Bruhat 阶次的格子双射

在本文中，我们引入了抽象字符串模块，并给出了抽象字符串模块的子模块格与相应抽象蛇图的完美匹配格之间的显式双射。特别是，我们明确了字符串模块的子模块与相应蛇图的完美匹配之间的直接对应关系。对于每个弦模，我们在对称群中定义一个 Coxeter 元，并在这些晶格和 Coxeter 元确定的弱 Bruhat 阶中的区间之间建立双射。利用字符串模块和蛇图之间的对应关系，我们给出了蛇图演算的新简明表述。