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Representation Theory

The representation theory faculty work on a range of problems related to Lie algebras and algebraic groups, using techniques drawn from combinatorics, geometry, and algebra. An algebraic group is an algebraic variety which is also a group and its tangent space at the identity comes with a Lie algebra structure. Representation theory is concerned with understanding how to embed the group (or the Lie algebra) into the set of matrices. Our research interests involve studying the rich collection of algebraic and geometric structures related to these embeddings, over the complex numbers and other fields. Such structures include Schubert varieties, affine and double affine Hecke algebras, symplectic reflection algebras, Springer fibers, nilpotent orbits, and affine Grassmannians, to name a few.

The Representation Theory Seminar is our main research venue, and most members also attend the Valley Geometry Seminar. In recent years, we have organized reading courses and special seminars geared specifically toward graduate students, including one on Kazhdan-Lusztig polynomials and another on cluster algebras.

The diagrams at the right are the connected Dynkin diagrams, which correspond to the simple Lie algebras over the complex numbers, a result of Cartan and Killing.

Research Areas

Algebraic combinatorics: Tom Braden, Eric Sommers

Algebraic groups: Tom Braden, Paul Gunnells, Jim Humphreys, Ivan Mirkovic, Eric Sommers

Geometric Langlands program: Ivan Mirkovic