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

İlke Çanakçı; Sibylle Schroll

doi:10.1016/j.aam.2020.102094

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

İlke Çanakçı
, Sibylle Schroll^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

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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.

Original language	English
Article number	102094
Pages (from-to)	1-22
Number of pages	22
Journal	Advances in Applied Mathematics
Volume	126
Early online date	17 Aug 2020
DOIs	https://doi.org/10.1016/j.aam.2020.102094
Publication status	Published - May 2021

Bibliographical note

Funding Information:
This work was supported through the Engineering and Physical Sciences Research Council , grant number EP/K026364/1 , UK. The first author was also supported by EPSRC through EP/P016014/1 and the second author by the EPSRC through an Early Career Fellowship EP/P016294/1 .

Publisher Copyright:
© 2020

Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

Funding

This work was supported through the Engineering and Physical Sciences Research Council , grant number EP/K026364/1 , UK. The first author was also supported by EPSRC through EP/P016014/1 and the second author by the EPSRC through an Early Career Fellowship EP/P016294/1 .

Keywords

Bruhat order
Distributive lattices
Perfect matchings
Snake graphs
String combinatorics
Symmetric groups

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10.1016/j.aam.2020.102094

Lattice bijections for string modules snake graphs and the weak Bruhat orderFinal published version, 435 KBLicence: Article 25fa Dutch Copyright Act

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KW - String combinatorics

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

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