Double dimer covers on snake graphs from super cluster expansions

Gregg Musiker; Nicholas Ovenhouse; Sylvester W Zhang

doi:10.1016/j.jalgebra.2022.05.033

Double dimer covers on snake graphs from super cluster expansions

Gregg Musiker, Nicholas Ovenhouse, Sylvester W Zhang

School of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ-lengths in a marked disk, generalizing Schiffler's T-path formula. In the present paper, we give an alternate combinatorial expression for these super λ-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.

Original language	English (US)
Pages (from-to)	325-381
Number of pages	57
Journal	Journal of Algebra
Volume	608
DOIs	https://doi.org/10.1016/j.jalgebra.2022.05.033
State	Published - Oct 15 2022

Bibliographical note

Funding Information:
The authors would like to thank the support of the NSF grants DMS-1745638 and DMS-1854162 , as well as the University of Minnesota UROP program. We would also like to thank Bruce Sagan for many helpful conversations and Ralf Schiffler for his questions motivating some of this work. Lastly, we thank the referee for their helpful edits to the paper.

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

Cluster algebras
Decorated super-Teichmüller spaces
Double dimers
Snake graphs

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10.1016/j.jalgebra.2022.05.033

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abstract = "In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ-lengths in a marked disk, generalizing Schiffler's T-path formula. In the present paper, we give an alternate combinatorial expression for these super λ-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.",

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AU - Ovenhouse, Nicholas

AU - Zhang, Sylvester W

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Y1 - 2022/10/15

N2 - In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ-lengths in a marked disk, generalizing Schiffler's T-path formula. In the present paper, we give an alternate combinatorial expression for these super λ-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.

AB - In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ-lengths in a marked disk, generalizing Schiffler's T-path formula. In the present paper, we give an alternate combinatorial expression for these super λ-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.

KW - Cluster algebras

KW - Decorated super-Teichmüller spaces

KW - Double dimers

KW - Snake graphs

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JO - Journal of Algebra

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ER -

Double dimer covers on snake graphs from super cluster expansions

Abstract

Bibliographical note

Keywords

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