Handbook

Representation theory combines the notions of symmetry and linearity, both of which are ubiquitous in mathematics. Initially, it utilised and unified ideas from group theory and linear algebra; however, it has progressed far beyond these humble beginnings. Representation theory is now known to have deep connections to other areas of mathematics and profound applications to the sciences and engineering. The unit starts by developing the theory of representations of finite groups over the complex numbers, with an emphasis on characters and symmetric groups. This theory forms the basis for the study of more advanced topics, such as the following examples:  applications to abstract group theory; representation-theoretic algorithms; random walks on groups; representations of Lie algebras; Schur-Weyl duality.