Symmetry, Structure, and Representations

This course provides a comprehensive introduction to the theory of finite groups, Lie groups, Lie algebras, and their representations.

We begin with the symmetric group and its representations, including Young–Frobenius theory and the modern Okounkov–Vershik approach, with an emphasis on the combinatorics of Young diagrams and Young tableaux.

After a concise review of smooth manifolds, we introduce Lie groups, their Lie algebras, and the action of a Lie group on a manifold together with the corresponding infinitesimal representations.

We then examine the structure of semisimple Lie algebras and develop the general theory of their finite-dimensional irreducible representations through the construction of highest weight modules. Further topics include the representation theory of the classical groups SL(N, ℂ) and SU(N), presented via the Schur–Weyl duality, and other key constructions in finite-dimensional representation theory.

Physical applications are interwoven throughout the course, such as the Wigner–Eckart theorem in quantum mechanics, Lorentz and Poincaré symmetries, the Dirac equation, and the classification of spinors in various space-time dimensions.

We conclude with symmetry-based methods for organizing the particle content of the Standard Model.