Lecture Notes on Geometric Crystals and Their Combinatorial Analogues

The first author would like to express his gratitude to Professor Okado for the opportunity to give the lectures, to Professor Kuniba for the great help in preparation of the lecture notes, and to RIMS for the hospitality. The lectures are based on the results of our research originated six years ago in [1] and continued in [2]. We start with the problem of birational Weyl group actions that served as the original motivation for this work (for the terminology and results on reductive algebraic groups, see, e.g., [8]).