12 Facts About Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood.

Representation theory is important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.

Representation theory is pervasive across fields of mathematics for two reasons.

The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces.

Representation theory therefore seeks to classify representations up to isomorphism.

Representation theory is notable for the number of branches it has, and the diversity of the approaches to studying representations of groups and algebras.

Major goal is to describe the "unitary dual", the space of irreducible unitary representations of G The theory is most well-developed in the case that G is a locally compact topological group and the representations are strongly continuous.

Harmonic analysis has been extended from the analysis of functions on a group G to functions on homogeneous spaces for G The theory is particularly well developed for symmetric spaces and provides a theory of automorphic forms .

Many of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.

Classically, the Representation theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group.

Important results in the Representation theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann–Roch theorem could be applied to calculate the dimension of the space of automorphic forms.