  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.
