  The tropical periodic Toda lattice (trop p-Toda) is a dynamical system
attracting attentions in the area of the interplay of integrable systems and
tropical geometry. We show that the Young diagrams associated with trop p-Toda
given by two very different definitions are identical. The first definition is
given by a Lax representation of the discrete periodic Toda lattice, and the
second one is associated with a generalization of the
Kerov-Kirillov-Reshetikhin bijection in the combinatorics of the Bethe ansatz.
By means of this identification, it is shown for the first time that the Young
diagrams given by the latter definition are preserved under time evolution.
This result is regarded as an important first step in clarifying the iso-level
set structure of this dynamical system in general cases, i.e. not restricted to
generic cases.
