R-systems

Pavel Galashin, Pavlo Pylyavskyy

Research output: Contribution to journalArticle

2 Scopus citations

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.

Original languageEnglish (US)
Article number22
JournalSelecta Mathematica, New Series
Volume25
Issue number2
DOIs
StatePublished - Jun 1 2019

Keywords

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

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