Among the finite-dimensional representations of quantum affine algebras, there exists a family of modules with desirable properties known as Kirillov-Reshetikhin (KR) modules. KR modules are parameterized by a vertex $r$ of the Dynkin diagram and a positive integer $l$, and they possess crystal structures in the sense of Kashiwara. Meanwhile, Berenstein and Kazhdan introduced the concept of "geometric crystals" as an algebraic-geometric analogue of crystals; it is conjectured that these exist in correspondence with sequences of KR modules where $r$ is fixed. Recently, we have obtained the "decoration functions" (in the sense of Berenstein and Kazhdan) for cases where the corresponding affine Lie algebra is of ADE type and $r$ corresponds to a minuscule weight representation. By applying ultradiscretization (tropicalization) to these geometric crystals with decoration functions, one obtains an integral convex polytope representation of the KR crystal—the crystal structure associated with the KR module—for any value of $l$.

In the talk, I will illustrate the construction procedures for crystals, geometric crystals, and integral convex polytopes using basic examples. For root systems of type A, these correspond respectively to (rectangular) Young tableaux, Grassmannians, and Gelfand-Tsetlin polytopes (corresponding to specific highest weight representations). This conjecture can be extended to all root systems, yet the area remains almost entirely unexplored for types other than type A.