Green-to-red sequences for positroids

Nicolas Ford, Khrystyna Serhiyenko

Research output: Contribution to journalArticlepeer-review

5 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 languageEnglish
Pages (from-to)164-182
Number of pages19
JournalJournal of Combinatorial Theory. Series A
Volume159
DOIs
StatePublished - Oct 2018

Bibliographical note

Funding Information:
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 .

Funding Information:
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.

Publisher Copyright:
© 2018 Elsevier Inc.

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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