From Wikipedia, the free encyclopedia.
In mathematics, a unitary representation of a group G is a linear representation of G on V, where the complex vector space V is a Hilbert space, and G acts by unitary transformations. Examples include symmetry groups of rotations in Euclidean space.
Unitary representations are closely connected with harmonic analysis; in the case of G abelian, as discussed by Pontryagin duality, can be most directly generalised to non-abelian groups by means of unitary representations. The classes of irreducible unitary representations of G makes up its unitary dual, a topological space which in many cases can be described. The general form of Plancherel theorem tries to describe the representation of G on L2(G) by means of a measure on the unitary dual. For G compact, this is done by the Peter-Weyl theorem; in that case the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree.
One of the pioneers in constructing a general theory of unitary representations was George Mackey. The theory is widely applied in quantum mechanics.
