Representation Theory

Representation theory is the branch of abstract algebra that studies how abstract groups act on concrete vector spaces. A representation of a group G is a homomorphism from G to the group of invertible linear transformations of a vector space V. The classification of representations — which groups have which representations, and what invariants distinguish them — is one of the deepest problems in modern mathematics.

The physical importance of representation theory is difficult to overstate. In quantum mechanics, physical states are vectors in a Hilbert space, and symmetries of the system are represented by unitary operators. The Standard Model of particle physics is, in large part, a catalog of the representations of the gauge group SU(3) × SU(2) × U(1). When a symmetry is spontaneously broken, the pattern of which representations acquire mass and which remain massless is controlled by the representation-theoretic structure of the symmetry group and its subgroups.

Representation theory connects group theory to linear algebra, abstract algebra to physics, and category theory to computation. It is the bridge between the abstract and the concrete.