# Representation Theory

Representation theory is the study of representations of algebraic objects by matrices. Given an algebraic object *A*, one looks at vector spaces *V* with functions *A*x*V* → *V* determining the map from *A* to the set of linear operators on *V*. This map is often asked to respect algebraic, analytic, topological, etc. properties of *A* and *V*, if there are any. Motivation, particular examples, and problems are of great importance to many parts of Mathematics as well as Physics. For instance, Unitary Symmetry, i.e. representations of *SU*(3), allow physicists to classify elementary particles.Representation theory research at Warwick centers around objects of Cartan-Dynkin-Weyl type, i.e., reductive algebraic groups, semisimple Lie algebra, finite groups of Lie type, quantum groups, Hecke algebras, Kac-Moody algebras etc. We are particularly interested in representations of affine Hecke algebras, Lusztig conjectures, and combinatorics related to crystal and canonical bases.

Representation theory is actively used by mathematicians working in many areas, for instance, soluble groups, homological algebra, K-theory, McKay correspondence, string theory, symplectic geometry.

Other areas of interest include:

- Representations of braid groups, Vassiliev invariants
- K-Theory
- McKay Correspondence
- Symplectic Geometry