Lattice bijections for string modules, snake graphs and the weak Bruhat order

Abstract

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. We then establish a bijection between the submodule lattice of the string module and the lattice given by 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.