Lattice bijections for string modules, snake graphs and the weak Bruhat order (Q2020038)

In this paper, the notion of abstract string module is introduced. Bijections between canonical submodule lattices of a string module, lattices given by an interval in the weak Bruhat order determined by the Coxeter element in a symmetric group associated to this string module, and perfect matching lattices of the corresponding snake graphs are provided. Some results on crossing snake graphs and their resolutions appearing in [\textit{İçakçı} and \textit{R. Schiffler}, J. Algebra 382, 240--281 (2013; Zbl 1319.13012)] are reviewed. In particular, a bijection between the perfect matching lattices of a pair of crossing snake graphs and the perfect matching lattices of their resolutions is presented in a simplified form.