Explore the intricate connections between Weyl polytopes, weights, and characters of highest weight modules in this comprehensive lecture. Delve into the fascinating world of algebraic and combinatorial methods in representation theory, focusing on infinite-dimensional Lie (super)algebras, quantum groups, and vertex algebras. Learn about the faces of Weyl polytopes and their relationship to weights and characters of highest weight modules. Gain insights from expert Apoorva Khare as part of the "Algebraic and Combinatorial Methods in Representation Theory" program organized by the International Centre for Theoretical Sciences. Enhance your understanding of modern representation theory and its applications in mathematics and physics during this 1 hour and 28 minutes presentation.