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)|
|Number of pages||43|
|Journal||Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions|
|State||Published - 2018|
Bibliographical noteFunding 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.
- Box ball system
- Discrete soliton
- Loop schur functions
- Rigged configuration