# Rigged configurations and cylindric loop Schur functions

### Thomas Lam

University of Michigan, Ann Arbor, USA### Pavlo Pylyavskyy

University of Minnesota, Minneapolis, USA### Reiho Sakamoto

Tokyo University of Science, Japan

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## 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.

## Cite this article

Thomas Lam, Pavlo Pylyavskyy, Reiho Sakamoto, Rigged configurations and cylindric loop Schur functions. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), no. 4, pp. 513–555

DOI 10.4171/AIHPD/61