## 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 language | English (US) |
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Pages (from-to) | 513-555 |

Number of pages | 43 |

Journal | Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions |

Volume | 5 |

Issue number | 4 |

DOIs | |

State | Published - 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