Representation theory

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

Let \(V\) be a vector space over a field \(\mathbb{F}\). For instance, suppose \(V\) is \(\R^n\) or \(\C^n\), 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 \(n\times n\) 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 \(n\times n\) matrices is a group under matrix multiplication, and the representation theory of groups analyzes a group by describing ("representing") its elements in terms of invertible matrices.
• Matrix addition and multiplication make the set of all \(n\times n\) matrices into an associative algebra, and hence there is a corresponding representation theory of associative algebras.
• If we replace matrix multiplication \(MN\) by the matrix commutator \(MN-NM\), then the \(n\times n\) matrices become instead a Lie algebra, leading to a representation theory of Lie algebras.

This generalizes to any field \(\mathbb{F}\) and any vector space \(V\) over \(\mathbb{F}\), with linear maps replacing matrices and composition replacing matrix multiplication: there is a group \(\text{GL}(V,\mathbb{F})\) of automorphisms of \(V\), an associative algebra \(\text{End}_{\mathbb{F}}(V)\) of all endomorphisms of \(V\), and a corresponding Lie algebra \(\mathfrak{gl}(V,\mathbb{F})\).

The vector space V is called the representation space of φ and its dimension (if finite) is called the dimension of the representation (sometimes degree, as in ). 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 (V,φ) 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 F, and hence recover a matrix representation with entries in the field F.

An effective or faithful representation is a representation (V,φ), for which the homomorphism φ is injective.

If \(V\) and \(W\) are vector spaces over \(\mathbb{F}\), equipped with representations \(\varphi\) and \(\psi\) of a group \(G\), then an equivariant map from \(V\) to \(W\) is a linear map \(\alpha:V\rightarrow W\) such that \[\alpha( g\cdot v ) = g \cdot \alpha(v)\] for all \(g\) in \(G\) and \(v\) in \(V\). In terms of \(\varphi:G\rightarrow\text{GL}(V)\) and \(\psi:G\rightarrow\text{GL}(W)\), this means \[\alpha\circ \varphi(g) = \psi(g)\circ \alpha\] for all \(g\) in \(G\), that is, the following diagram commutes:

Equivariant maps for representations of an associative or Lie algebra are defined similarly. If \(\alpha\) is invertible, then it is said to be an isomorphism, in which case \(V\) and \(W\) (or, more precisely, \(\varphi\) and \(\psi\)) 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 \(G\), it is on occasion called a \(G\)-map or \(G\)-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.

If \((V,\psi)\) is a representation of (say) a group \(G\), and \(W\) is a linear subspace of \(V\) that is preserved by the action of \(G\) in the sense that for all \(w \in W\) and \(g\in G\), \(\psi(g)\cdot w \in W\) (Serre calls these \(W\) stable under \(G\)), then \(W\) is called a subrepresentation: by defining \[\phi:G \to \text{Aut}(W)\] where \(\phi(g)\) is the restriction of \(\psi(g)\) to \(W\), \((W,\phi)\) is a representation of \(G\) and the inclusion of \(W \hookrightarrow V\) is an equivariant map. The quotient space \(V/W\) can also be made into a representation of \(G\). If \(V\) has exactly two subrepresentations, namely the trivial subspace {0} and \(V\) itself, then the representation is said to be irreducible; if \(V\) 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 \[\alpha: (V,\psi) \to (V',\psi')\] between irreducible representations is either the zero map or an isomorphism, since its kernel and image are subrepresentations. In particular, when \(V = V'\), this shows that the equivariant endomorphisms of \(V\) 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 \(V\) is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if \(V\) 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 \((\mathbb{R},+)\) has a two dimensional representation \[\phi(a) = \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}\] This group has the vector \(\begin{bmatrix} 1 & 0 \end{bmatrix}^\mathsf{T}\) fixed by this homomorphism, but the complement subspace maps to \[\begin{bmatrix} 0 \\ 1 \end{bmatrix} \mapsto \begin{bmatrix} a \\ 1 \end{bmatrix}\] giving only one irreducible subrepresentation. This is true for all unipotent groups.

If (V,φ) and (W,ψ) are representations of (say) a group G, then the direct sum of V and W is a representation, in a canonical way, via the equation

\(g\cdot (v,w) = (g\cdot v, g\cdot w).\)

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.

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 (at least over fields of characteristic zero) include finite groups (see Maschke's theorem), 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.

Suppose \(\phi_1:G\rightarrow \mathrm{GL}(V_1)\) and \(\phi_2:G\rightarrow \mathrm{GL}(V_2)\) are representations of a group \(G\). Then we can form a representation \(\phi_1\otimes\phi_2\) of G acting on the tensor product vector space \(V_1\otimes V_2\) as follows:

\((\phi_1\otimes\phi_2)(g)=\phi_1(g)\otimes\phi_2(g)\).

If \(\phi_1\) and \(\phi_2\) are representations of a Lie algebra, then the correct formula to use is

\((\phi_1\otimes\phi_2)(X)=\phi_1(X)\otimes I+I\otimes\phi_2(X)\).

This product can be recognized as the coproduct on a coalgebra. In general, the tensor product of irreducible representations is not irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations is known as Clebsch-Gordan theory.

In the case of the representation theory of the group SU(2) (or equivalently, of its complexified Lie algebra \(\mathrm{sl}(2;\mathbb{C})\)), the decomposition is easy to work out. The irreducible representations are labeled by a parameter \(l\) that is a non-negative integer or half integer; the representation then has dimension \(2l+1\). Suppose we take the tensor product of the representation of two representations, with labels \(l_1\) and \(l_2,\) where we assume \(l_1\geq l_2\). Then the tensor product decomposes as a direct sum of one copy of each representation with label \(l\), where \(l\) ranges from \(l_1-l_2\) to \(l_1+l_2\) in increments of 1. If, for example, \(l_1=l_2=1\), then the values of \(l\) that occur are 0, 1, and 2. Thus, the tensor product representation of dimension \((2l_1+1) \times (2l_2+1) = 3\times 3=9\) decomposes as a direct sum of a 1-dimensional representation \((l=0),\) a 3-dimensional representation \((l=1),\) and a 5-dimensional representation \((l=2)\).

Representation theory is notable for the number of branches it has, and the diversity of the approaches to studying representations of groups and algebras. Although, all the theories have in common the basic concepts discussed already, they differ considerably in detail. The differences are at least 3-fold:

1. Representation theory depends upon the type of algebraic object being represented. There are several different classes of groups, associative algebras and Lie algebras, and their representation theories all have an individual flavour.
2. Representation theory depends upon the nature of the vector space on which the algebraic object is represented. The most important distinction is between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (for example, whether or not the space is a Hilbert space, Banach space, etc.). Additional algebraic structures can also be imposed in the finite-dimensional case.
3. Representation theory depends upon the type of field over which the vector space is defined. The most important cases are the field of complex numbers, the field of real numbers, finite fields, and fields of p-adic numbers. Additional difficulties arise for fields of positive characteristic and for fields that are not algebraically closed.

Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to geometry and crystallography. Representations of finite groups exhibit many of the features of the general theory and point the way to other branches and topics in representation theory.

Over a field of characteristic zero, the representation of a finite group G has a number of convenient properties. First, the representations of G are semisimple (completely reducible). This is a consequence of Maschke's theorem, which states that any subrepresentation V of a G-representation W has a G-invariant complement. One proof is to choose any projection π from W to V and replace it by its average π_G defined by

\(\pi_G(x) = \frac1{|G|}\sum_{g\in G} g\cdot \pi(g^{-1}\cdot x).\)

π_G is equivariant, and its kernel is the required complement.

The finite-dimensional G-representations can be understood using character theory: the character of a representation φ: G → GL(V) is the class function χ_φ: G → F defined by

\(\chi_{\varphi}(g) = \mathrm{Tr}(\varphi(g))\)

where \(\mathrm{Tr}\) is the trace. An irreducible representation of G is completely determined by its character.

Maschke's theorem holds more generally for fields of positive characteristic p, such as the finite fields, as long as the prime p is coprime to the order of G. When p and |G| have a common factor, there are G-representations that are not semisimple, which are studied in a subbranch called modular representation theory.

Averaging techniques also show that if F is the real or complex numbers, then any G-representation preserves an inner product \(\langle\cdot,\cdot\rangle\) on V in the sense that

\(\langle g\cdot v,g\cdot w\rangle = \langle v,w\rangle\)

Condensed: the full section is in Wikipedia.

Modular representations of a finite group G are representations over a field whose characteristic is not coprime to |G|, so that Maschke's theorem no longer holds (because |G| is not invertible in F and so one cannot divide by it). Nevertheless, Richard Brauer extended much of character theory to modular representations, and this theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were "too small".

As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.

A unitary representation of a group G is a linear representation φ of G on a real or (usually) complex Hilbert space V such that φ(g) is a unitary operator for every g ∈ G. Such representations have been widely applied in quantum mechanics since the 1920s, thanks in particular to the influence of Hermann Weyl, and this has inspired the development of the theory, most notably through the analysis of representations of the Poincaré group by Eugene Wigner. One of the pioneers in constructing a general theory of unitary representations (for any group G rather than just for particular groups useful in applications) was George Mackey, and an extensive theory was developed by Harish-Chandra and others in the 1950s and 1960s.

A 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 (Hausdorff) topological group and the representations are strongly continuous. For G abelian, the unitary dual is just the space of characters, while for G compact, the Peter-Weyl theorem shows that the irreducible unitary representations are finite-dimensional and the unitary dual is discrete. For example, if G is the circle group S, then the characters are given by integers, and the unitary dual is Z.

For non-compact G, the question of which representations are unitary is a subtle one. Although irreducible unitary representations must be "admissible" (as Harish-Chandra modules) and it is easy to detect which admissible representations have a nondegenerate invariant sesquilinear form, it is hard to determine when this form is positive definite. An effective description of the unitary dual, even for relatively well-behaved groups such as real reductive Lie groups (discussed below), remains an important open problem in representation theory. It has been solved for many particular groups, such as SL(2,R) and the Lorentz group.

The duality between the circle group S and the integers Z, or more generally, between a torus T and Z is well known in analysis as the theory of Fourier series, and the Fourier transform similarly expresses the fact that the space of characters on a real vector space is the dual vector space. Thus unitary representation theory and harmonic analysis are intimately related, and abstract harmonic analysis exploits this relationship, by developing the analysis of functions on locally compact topological groups and related spaces.

A major goal is to provide a general form of the Fourier transform and the Plancherel theorem. This is done by constructing a measure on the unitary dual and an isomorphism between the regular representation of G on the space L(G) of square integrable functions on G and its representation on the space of L functions on the unitary dual. Pontrjagin duality and the Peter-Weyl theorem achieve this for abelian and compact G respectively.

Another approach involves considering all unitary representations, not just the irreducible ones. These form a category, and Tannaka-Krein duality provides a way to recover a compact group from its category of unitary representations.

If the group is neither abelian nor compact, no general theory is known with an analogue of the Plancherel theorem or Fourier inversion, although Alexander Grothendieck extended Tannaka-Krein duality to a relationship between linear algebraic groups and tannakian categories.

Harmonic analysis has also 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 (discussed below).

A Lie group is a group that is also a smooth manifold. Many classical groups of matrices over the real or complex numbers are Lie groups. 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.

The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply. This theory can be extended to finite-dimensional representations of semisimple Lie groups using Weyl's unitary trick: each semisimple real Lie group G has a complexification, which is a complex Lie group G, and this complex Lie group has a maximal compact subgroup K. The finite-dimensional representations of G closely correspond to those of K.

A general Lie group is a semidirect product of a solvable Lie group and a semisimple Lie group (the Levi decomposition). The classification of representations of solvable Lie groups is intractable in general, but often easy in practical cases. Representations of semidirect products can then be analysed by means of general results called Mackey theory, which is a generalization of the methods used in Wigner's classification of representations of the Poincaré group.