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. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations.
The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these 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. Representations of more abstract objects in terms of familiar linear algebra can elucidate properties and simplify calculations within more abstract theories. 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 many 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.
The success of representation theory has led to numerous generalizations. One of the most general is in category theory. 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. This description points to two natural generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.

Definitions and concepts

Let be a vector space over a field. For instance, suppose is or, the standard n-dimensional space of column vectors over the real or complex numbers, respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using matrices of real or complex numbers.
There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.

The set of all invertible matrices is a group under matrix multiplication, and the representation theory of groups analyzes a group by describing its elements in terms of invertible matrices.
Matrix addition and multiplication make the set of all matrices into an associative algebra, and hence there is a corresponding representation theory of associative algebras.
If we replace matrix multiplication by the matrix commutator, then the matrices become instead a Lie algebra, leading to a representation theory of Lie algebras.

This generalizes to any field and any vector space over, with linear maps replacing matrices and composition replacing matrix multiplication: there is a group of automorphisms of, an associative algebra of all endomorphisms of, and a corresponding Lie algebra.

Definition

Action

There are two ways to define a representation. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication.
A representation of a group or algebra on a vector space is a map
with two properties.

For any in , the map
is linear.
If we introduce the notation g · v for , then for any g₁, g₂ in G and v in V:
where e is the identity element of G and g₁g₂ is the group product in G.

The definition for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation is omitted. Equation is an abstract expression of the associativity of matrix multiplication. This doesn't hold for the matrix commutator and also there is no identity element for the commutator. Hence for Lie algebras, the only requirement is that for any x₁, x₂ in A and v in V:
where is the Lie bracket, which generalizes the matrix commutator MN − NM.

Mapping

The second way to define a representation focuses on the map φ sending g in G to a linear map φ: V → V, which satisfies
and similarly in the other cases. This approach is both more concise and more abstract.
From this point of view:

a representation of a group G on a vector space V is a group homomorphism φ: G → GL;
a representation of an associative algebra A on a vector space V is an algebra homomorphism φ: A → End_F;
a representation of a Lie algebra on a vector space is a Lie algebra homomorphism.
Terminology

The vector space V is called the representation space of φ and its dimension is called the dimension of the representation. It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context; otherwise the notation can be used to denote a representation.
When V is of finite dimension n, one can choose a basis for V to identify V with Fⁿ, and hence recover a matrix representation with entries in the field F.
An effective or faithful representation is a representation, for which the homomorphism φ is injective.

Equivariant maps and isomorphisms

If and are vector spaces over, equipped with representations and of a group, then an equivariant map from to is a linear map such that
for all in and in. In terms of and, this means
for all in, that is, the following diagram commutes:
Equivariant maps for representations of an associative or Lie algebra are defined similarly. If is invertible, then it is said to be an isomorphism, in which case and are isomorphic representations, also phrased as equivalent representations. An equivariant map is often called an intertwining map of representations. Also, in the case of a group, it is on occasion called a -map or -linear map.
Isomorphic representations are, for practical purposes, "the same"; they provide the same information about the group or algebra being represented. Representation theory therefore seeks to classify representations up to isomorphism.

Subrepresentations, quotients, and irreducible representations

If is a representation of a group, and is a linear subspace of that is preserved by the action of in the sense that for all and, , then is called a subrepresentation: by defining where is the restriction of to, is a representation of and the inclusion of is an equivariant map. The quotient space can also be made into a representation of. If has exactly two subrepresentations, namely the trivial subspace and itself, then the representation is said to be irreducible; if has a proper nontrivial subrepresentation, the representation is said to be reducible.
The definition of an irreducible representation implies Schur's lemma: an equivariant map between irreducible representations is either the zero map or an isomorphism, since its kernel and image are subrepresentations. In particular, when, this shows that the equivariant endomorphisms of form an associative division algebra over the underlying field F. If F is algebraically closed, the only equivariant endomorphisms of an irreducible representation are the scalar multiples of the identity.
Irreducible representations are the building blocks of representation theory for many groups: if a representation is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if is finite-dimensional, then both the subrepresentation and the quotient have smaller dimension. There are counterexamples where a representation has a subrepresentation, but only has one non-trivial irreducible component. For example, the additive group has a two dimensional representation
This group has the vector fixed by this homomorphism, but the complement subspace maps to
giving only one irreducible subrepresentation. This is true for all unipotent groups.

Direct sums and indecomposable representations

If and are representations of a group G, then the direct sum of V and W is a representation, in a canonical way, via the equation
The direct sum of two representations carries no more information about the group G than the two representations do individually. If a representation is the direct sum of two proper nontrivial subrepresentations, it is said to be decomposable. Otherwise, it is said to be indecomposable.

Complete reducibility

In favorable circumstances, every finite-dimensional representation is a direct sum of irreducible representations: such representations are said to be semisimple. In this case, it suffices to understand only the irreducible representations. Examples where this "complete reducibility" phenomenon occurs include finite groups, compact groups, and semisimple Lie algebras.
In cases where complete reducibility does not hold, one must understand how indecomposable representations can be built from irreducible representations by using extensions of quotients by subrepresentations.