Rigged configurations and cylindric loop Schur functions. (English) Zbl 1400.05267
Summary: 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.
MSC:
05E10 | Combinatorial aspects of representation theory |
37F20 | Combinatorics and topology in relation with holomorphic dynamical systems |
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |