Rigged Configurations and Cylindric Loop Schur Functions

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.

32 pages. (v2) Corrections on Section 6.3, Conjecture 5.3 and other typos. Acknowledgments added. (v3) Final version with minor revision at Lemma 6.13