Representation theory is fundamental in the study of objects with symmetry. It arises in contexts as diverse as card shuffling and quantum mechanics. An early success was the work of Schur and Weyl, who computed the representation theory of the symmetric and unitary groups; the answer is closely related to the classical theory of symmetric functions and deeper study leads to intricate questions in combinatorics.
More recently, methods from geometry and topology have greatly enhanced our understanding of these questions (“geometric representation theory”). The study of affine Lie algebras and quantum groups has brought many new ideas and viewpoints, and representation theory now furnishes a basic language for other fields, including the modern theory of automorphic forms.
All of these aspects are studied by Stanford faculty. Topics of recent seminars include combinatorial representation theory as well as quantum groups.