Ultradiscrete Soliton Systems and Combinatorial Representation Theory

This lecture note is intended to be a brief introduction to a recent development on the interplay between the ultradiscrete (or tropical) soliton systems and the combinatorial representation theory. We will concentrate on the simplest cases which admit elementary explanations without losing essential ideas of the theory. In particular we give definitions for the main constructions corresponding to the vector representation of type $A_1^{(1)}$.