Green-to-red sequences for positroids

Nicolas Ford; Khrystyna Serhiyenko

doi:10.1016/j.jcta.2018.06.001

Green-to-red sequences for positroids

Nicolas Ford, Khrystyna Serhiyenko

Mathematics

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

7 Scopus citations

Abstract

L-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each L-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of the corresponding positroid variety. We study quivers arising from these diagrams and show that they can be constructed from the well-behaved quivers associated to Grassmannians by deleting and merging certain vertices. Then, we prove that quivers coming from arbitrary L-diagrams, and more generally reduced plabic graphs, admit a particular sequence of mutations called a green-to-red sequence.

Original language	English
Pages (from-to)	164-182
Number of pages	19
Journal	Journal of Combinatorial Theory. Series A
Volume	159
DOIs	https://doi.org/10.1016/j.jcta.2018.06.001
State	Published - Oct 2018

Bibliographical note

Publisher Copyright:
© 2018 Elsevier Inc.

Funding

Acknowledgments: The authors would like to thank Lauren Williams for bringing this project to their attention, Greg Muller for several fruitful conversations, and Steven Karp for valuable comments on the exposition. The second author was supported by the NSF Postdoctoral fellowship MSPRF-1502881.

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	MSPRF-1502881

Keywords

Green-to-red sequence
Le-diagram
Plabic graph
Positroid
Quiver mutation

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jcta.2018.06.001

Cite this

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title = "Green-to-red sequences for positroids",

abstract = "L-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each L-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of the corresponding positroid variety. We study quivers arising from these diagrams and show that they can be constructed from the well-behaved quivers associated to Grassmannians by deleting and merging certain vertices. Then, we prove that quivers coming from arbitrary L-diagrams, and more generally reduced plabic graphs, admit a particular sequence of mutations called a green-to-red sequence.",

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author = "Nicolas Ford and Khrystyna Serhiyenko",

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year = "2018",

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doi = "10.1016/j.jcta.2018.06.001",

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AB - L-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each L-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of the corresponding positroid variety. We study quivers arising from these diagrams and show that they can be constructed from the well-behaved quivers associated to Grassmannians by deleting and merging certain vertices. Then, we prove that quivers coming from arbitrary L-diagrams, and more generally reduced plabic graphs, admit a particular sequence of mutations called a green-to-red sequence.

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KW - Plabic graph

KW - Positroid

KW - Quiver mutation

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ER -

Green-to-red sequences for positroids

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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