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Annales de l’Institut Henri Poincaré D

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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