Representation theory

Lore & Background

Representation theory reduces problems in abstract algebra to problems in linear algebra, a well-understood subject. Representations of abstract objects in terms of familiar linear algebra can elucidate properties and simplify calculations within more abstract theories. For example, representing a group by an infinite-dimensional Hilbert space allows methods of analysis to be applied to group theory.

Reader's Guide

Representation theory is pervasive across fields of mathematics. Its applications are diverse: it generalizes Fourier analysis via harmonic analysis, is connected to geometry through invariant theory and the Erlangen program, and impacts number theory via automorphic forms and the Langlands program. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics, and topology. The success of representation theory has led to generalizations in category theory, where algebraic objects can be viewed as categories and representations as functors to the category of vector spaces. This points to two natural generalizations: replacing algebraic objects with more general categories, and replacing the target category of vector spaces with other well-understood categories.

Did You Know?

• Representation theory can describe how the symmetry group of a physical system affects the solutions of equations describing that system.
• The set of all invertible n×n matrices forms a group under matrix multiplication, and representation theory analyzes a group by describing its elements in terms of such matrices.
• If matrix multiplication is replaced by the matrix commutator MN−NM, the n×n matrices become a Lie algebra, leading to representation theory of Lie algebras.
• Representation theory generalizes Fourier analysis via harmonic analysis.