Inequalities Defining Polyhedral Realizations and Monomial Realizations in Crystal Bases

Watch a 54-minute mathematics lecture exploring the relationship between polyhedral realizations and monomial realizations in crystal base theory. Dive into the study of crystal bases B(∞) and B(λ) as fundamental tools for understanding representations of Lie algebras and quantum groups. Learn how these bases provide crucial information about integrable highest weight representations and Verma modules through combinatorial descriptions. Examine the Nakashima-Zelevinsky polyhedral realizations of B(∞) as integer points in convex cones, and understand their connection to string cones in finite dimensional simple Lie algebras. Explore Kashiwara and Nakajima's monomial realizations as combinatorial expressions of crystal bases B(λ) using Laurent monomials. Consider a proposed conjecture linking the inequalities that define polyhedral realization cones to monomial realizations of fundamental representations. Presented by Yuki Kanakubo from Ibaraki University at the Institut des Hautes Etudes Scientifiques (IHES).