Rigged configurations and cylindric loop schur functions

Thomas Lam, Pavlo Pylyavskyy, Reiho Sakamoto

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Rigged configurations are known to provide action-angle variables for remarkable discrete dynamical systems known as box-ball systems. We conjecture an explicit piecewise-linear formula to obtain the shapes of a rigged configuration from a tensor product of one-row crystals. We introduce cylindric loop Schur functions and show that they are invariants of the geometric R-matrix. Our piecewise-linear formula is obtained as the tropicalization of ratios of cylindric loop Schur functions. We prove our conjecture for the first shape of a rigged configuration, thus giving a piecewise-linear formula for the lengths of the solitons of a box-ball system.

Original languageEnglish (US)
Pages (from-to)513-555
Number of pages43
JournalAnnales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions
Volume5
Issue number4
DOIs
StatePublished - 2018

Bibliographical note

Funding Information:
Thomas Lam was supported by NSF grants DMS-1160726,. DMS-1464693, and a Simons Fellowship. Pavlo Pylyavskyy was supported by NSF grants DMS-1068169, DMS-1148634, DMS-1351590 and a Sloan Fellowship.

Keywords

  • Box ball system
  • Discrete soliton
  • Loop schur functions
  • Rigged configuration
  • Tropicalization

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