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

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Bijection Bijection, injection and surjection Bruhat order Coxeter element Mathematics Combinatorics Coxeter group Lattice (music) Discrete mathematics Physics Acoustics

̛İlke Çanakçı , Sibylle Schroll ·

YOU? · · 2018 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1811.06064 · OA: W2900651975

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.