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. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.
Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. For instance, representing a group by an infinite-dimensional Hilbert space allows methods of analysis to be applied to the theory of groups. Furthermore, representation theory is important in physics because it can describe how the symmetry group of a physical system affects the solutions of equations describing that system.
Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse. In addition to its impact on algebra, representation theory
generalizes Fourier analysis via harmonic analysis,
is connected to geometry via invariant theory and the Erlangen program,
has an impact in number theory via automorphic forms and the Langlands program.
There are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.
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In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X.
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N, i.e. φ is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on M.
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
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AMER MATHEMATICAL SOC2022
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