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R-systems

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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.

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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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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Pavel Galashin

  2. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55414, USA

    Pavlo Pylyavskyy

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  1. Pavel Galashin
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  2. Pavlo Pylyavskyy
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Correspondence to Pavel Galashin.

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Galashin, P., Pylyavskyy, P. R-systems. Sel. Math. New Ser. 25, 22 (2019). https://doi.org/10.1007/s00029-019-0470-2

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