Rigged configurations and cylindric loop Schur functions
Annales de l’Institut Henri Poincaré D, Tome 5 (2018) no. 4, pp. 513-555.

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.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/61
Classification : 05-XX, 37-XX, 70-XX
Mots-clés : Rigged configuration, discrete soliton, box ball system, tropicalization, loop schur functions 
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Lam, Thomas; Pylyavskyy, Pavlo; Sakamoto, Reiho. Rigged configurations and cylindric loop Schur functions. Annales de l’Institut Henri Poincaré D, Tome 5 (2018) no. 4, pp. 513-555. doi : 10.4171/aihpd/61. https://www.numdam.org/articles/10.4171/aihpd/61/
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  • Frieden, Gabriel The geometric R-matrix for affine crystals of type A, Advances in Mathematics, Volume 392 (2021), p. 108000 | DOI:10.1016/j.aim.2021.108000
  • Scrimshaw, Travis Uniform description of the rigged configuration bijection, Selecta Mathematica, Volume 26 (2020) no. 3 | DOI:10.1007/s00029-020-00564-8

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