Explore the theory of crystal bases and their polyhedral realizations in this 44-minute lecture from the Institut des Hautes Etudes Scientifiques (IHES). Delve into the representation theory of Lie algebras and quantum groups, focusing on how crystal bases can be realized as combinatorial objects to reveal skeletal structures of representations. Learn about Nakashima and Zelevinsky's "polyhedral realizations" and the challenge of finding explicit forms of inequalities that define polyhedral convex cones and polytopes. Gain insights into the theory of Lie algebras and quantum groups, with a specific focus on classical affine types. Discover how extended Young diagrams are used to express explicit forms of inequalities in these cases. This talk by Yuki Kanabuko from the Max Planck Institute for Mathematics (MPIM) offers a deep dive into advanced mathematical concepts at the intersection of algebra, geometry, and combinatorics.