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

TL;DR

This paper establishes a novel correspondence between submodule lattices of string modules, perfect matchings of snake graphs, and intervals in the weak Bruhat order, providing new insights into their combinatorial structures.

Contribution

It introduces abstract string modules, constructs explicit bijections with snake graph matchings, and relates these to Coxeter elements and the weak Bruhat order.

Findings

Bijection between submodule lattice and perfect matching lattice

Explicit correspondence between string modules and snake graphs

New formulation of snake graph calculus

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, 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.