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R-systems

  • Pavel GalashinEmail author
  • Pavlo Pylyavskyy
Article
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Abstract

Birational toggling on Gelfand–Tsetlin patterns appeared first in the study of geometric crystals and geometric Robinson–Schensted–Knuth correspondence. Based on these birational toggle relations, Einstein and Propp introduced a discrete dynamical system called birational rowmotion associated with a partially ordered set. We generalize birational rowmotion to the class of arbitrary strongly connected directed graphs, calling the resulting discrete dynamical system the R-system. We study its integrability from the points of view of singularity confinement and algebraic entropy. We show that in many cases, singularity confinement in an R-system reduces to the Laurent phenomenon either in a cluster algebra, or in a Laurent phenomenon algebra, or beyond both of those generalities, giving rise to many new sequences with the Laurent property possessing rich groups of symmetries. Some special cases of R-systems reduce to Somos and Gale-Robinson sequences.

Keywords

Birational rowmotion Toggle Laurent phenomenon Cluster algebra Singularity confinement Algebraic entropy Arborescence Superpotential 

Mathematics Subject Classification

Primary 37K10 Secondary 13F60 05E99 

Notes

Acknowledgements

The material in this section is largely based on conversations with Thomas Lam. We are grateful to him for introducing us to this beautiful subject. We also thank Alex Postnikov and Steven Karp for related discussions. Finally, we are indebted to the anonymous referee for their careful reading of the first version of the manuscript.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsUniversity of MinnesotaMinneapolisUSA

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