The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Annales de l’Institut Henri Poincaré D


Full-Text PDF (415 KB) | Metadata | Table of Contents | AIHPD summary
Volume 5, Issue 4, 2018, pp. 513–555
DOI: 10.4171/AIHPD/61

Published online: 2018-07-25

Rigged configurations and cylindric loop Schur functions

Thomas Lam[1], Pavlo Pylyavskyy[2] and Reiho Sakamoto[3]

(1) University of Michigan, Ann Arbor, USA
(2) University of Minnesota, Minneapolis, USA
(3) Tokyo University of Science, Japan

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.

Keywords: Rigged configuration, discrete soliton, box ball system, tropicalization, loop schur functions

Lam Thomas, Pylyavskyy Pavlo, Sakamoto Reiho: Rigged configurations and cylindric loop Schur functions. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), 513-555. doi: 10.4171/AIHPD/61