View source for Representation theory

{{short description|Branch of mathematics that studies abstract algebraic structures}}
{{otheruses4|the theory of representations of algebraic structures by linear transformations and matrices|representation theory in other disciplines|Representation (disambiguation)}}{{confused|Group presentation}}

[[File:hexagon reflections.svg|thumb|right|250px|Representation theory studies how algebraic structures "act" on objects. A simple example is the way a polygon is transformed by [[Dihedral group|its symmetries under reflections and rotations]], which are all [[linear transformations]] about the center of the polygon.]]

'''Representation theory''' is a branch of [[mathematics]] that studies [[abstract algebra|abstract]] [[algebraic structure]]s by ''representing'' their [[element (set theory)|elements]] as [[linear transformation]]s of [[vector space]]s, and studies [[Module (mathematics)|modules]] over these abstract algebraic structures.<ref>Classic texts on representation theory include {{Harvtxt|Curtis|Reiner|1962}} and {{Harvtxt|Serre|1977}}. Other excellent sources are {{Harvtxt|Fulton|Harris|1991}} and {{Harvtxt|Goodman|Wallach|1998}}.</ref><ref>{{Cite web|url=https://ncatlab.org/nlab/show/representation+theory|title=representation theory in nLab|website=ncatlab.org|access-date=2019-12-09}}</ref> In essence, a representation makes an abstract algebraic object more concrete by describing its elements by [[matrix (mathematics)|matrices]] and their [[algebraic operation]]s (for example, [[matrix addition]], [[matrix multiplication]]).

The [[algebra]]ic objects amenable to such a description include [[group (mathematics)|groups]], [[associative algebra]]s and [[Lie algebra]]s. The most prominent of these (and historically the first) is the [[group representation|representation theory of groups]], in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.<ref>For the history of the representation theory of finite groups, see {{Harvtxt|Lam|1998}}. For algebraic and Lie groups, see {{Harvtxt|Borel|2001}}.</ref><ref name=":0">{{Cite web|url=http://www-math.mit.edu/~etingof/replect.pdf|title=Introduction to representation theory|last1=Etingof|first1=Pavel|last2=Golberg|first2=Oleg|date=January 10, 2011|website=www-math.mit.edu|access-date=2019-12-09 |last3=Hensel |first3=Sebastian |last4=Liu |first4=Tiankai |last5=Schwendner |first5=Alex|last6=Vaintrob|first6=Dmitry|last7=Yudovina|first7=Elena}}</ref>

Representation theory is a useful method because it reduces problems in [[abstract algebra]] to problems in [[linear algebra]], a subject that is well understood.<ref>{{cite web |last1=Ronan |first1=Mark Andrew |title=linear algebra |url=https://www.britannica.com/science/linear-algebra |website=Encyclopedia Britannica |access-date=8 July 2024 | quote = linear algebra is very well understood}}</ref><ref name=linalg> There are many textbooks on [[vector spaces]] and [[linear algebra]]. For an advanced treatment, see {{Harvtxt|Kostrikin|Manin|1997}}.</ref> 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 [[mathematical analysis|analysis]] to be applied to the theory of groups.<ref>{{Harvnb|Sally|Vogan|1989}}.</ref><ref name=":1">{{Cite web|url=https://math.berkeley.edu/~teleman/math/RepThry.pdf|title=Representation Theory|last=Teleman|first=Constantin|date=2005|website=math.berkeley.edu|access-date=2019-12-09}}</ref> 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.<ref name=Sternberg>{{Harvnb|Sternberg|1994}}.</ref>

Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse.<ref>{{Harvnb|Lam|1998|p=372}}.</ref> In addition to its impact on algebra, representation theory
* generalizes [[Fourier analysis]] via [[harmonic analysis]],<ref name=Folland>{{Harvnb|Folland|1995}}.</ref> 
* is connected to [[geometry]] via [[invariant theory]] and the [[Erlangen program]],<ref>{{Harvnb|Goodman|Wallach|1998}}, {{Harvnb|Olver|1999}}, {{Harvnb|Sharpe|1997}}.</ref>
* has an impact in number theory via [[automorphic form]]s and the [[Langlands program]].<ref>{{Harvnb|Borel|Casselman|1979}}, {{Harvnb|Gelbart|1984}}.</ref>

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]].<ref>See the previous footnotes.</ref>

The success of representation theory has led to numerous generalizations. One of the most general is in [[category theory]].<ref name=SSA>{{Harvnb|Simson|Skowronski|Assem|2007}}.</ref> The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as [[functor]]s from the object category to the [[category of vector spaces]].<ref name=":0" /> 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 <math>V</math> be a [[vector space]] over a [[field (mathematics)|field]] <math>\mathbb{F}</math>.<ref name=linalg/> For instance, suppose <math>V</math> is <math>\R^n</math> or <math>\C^n</math>, the standard ''n''-dimensional space of [[column vector]]s over the [[real number|real]] or [[complex number]]s, respectively. In this case, the idea of representation theory is to do [[abstract algebra]] concretely by using <math>n\times n</math> [[matrix (mathematics)|matrices]] of real or complex numbers.

There are three main sorts of [[algebra]]ic objects for which this can be done: [[group (mathematics)|groups]], [[associative algebra]]s and [[Lie algebra]]s.<ref>{{Harvnb|Fulton|Harris|1991}}, {{Harvnb|Simson|Skowronski|Assem|2007}}, {{Harvnb|Humphreys|1972a}}.</ref><ref name=":0" />

* The set of all ''[[invertible matrix|invertible]]'' <math>n\times n</math> matrices is a group under [[matrix multiplication]], and the [[group representation|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'' <math>n\times n</math> matrices into an associative algebra, and hence there is a corresponding [[representation of an associative algebra|representation theory of associative algebras]].
* If we replace matrix multiplication <math>MN</math> by the matrix [[commutator]] <math>MN-NM</math>, then the <math>n\times n</math> matrices become instead a Lie algebra, leading to a [[representation of a Lie algebra|representation theory of Lie algebras]].

This generalizes to any field <math>\mathbb{F}</math> and any vector space <math>V</math> over <math>\mathbb{F}</math>, with [[linear map]]s replacing matrices and [[function composition|composition]] replacing matrix multiplication: there is [[general linear group#General linear group of a vector space|a group]] <math>\text{GL}(V,\mathbb{F})</math> of [[linear map#Endomorphisms and automorphisms|automorphism]]s of <math>V</math>, an associative algebra <math>\text{End}_{\mathbb{F}}(V)</math> of all endomorphisms of <math>V</math>, and a corresponding Lie algebra <math>\mathfrak{gl}(V,\mathbb{F})</math>.

===Definition===
{{See also|group representation|algebra representation|Lie algebra representation}}
====Action====
There are two ways to define a representation.<ref>This material can be found in standard textbooks, such as {{Harvtxt|Curtis|Reiner|1962}}, {{Harvtxt|Fulton|Harris|1991}}, {{Harvtxt|Goodman|Wallach|1998}}, {{Harvtxt|James|Liebeck|1993}}, {{Harvtxt|Humphreys|1972a}}, {{Harvtxt|Jantzen|2003}}, {{Harvtxt|Knapp|2001}} and {{Harvtxt|Serre|1977}}.</ref> The first uses the idea of an [[Group action (mathematics)|action]], generalizing the way that matrices act on column vectors by matrix multiplication. 

A ''representation'' of a [[group (mathematics)|group]] <math>G</math> or (associative or Lie) algebra <math>A</math> on a vector space <math>V</math> is a map
<math display="block"> \Phi\colon G\times V \to V \quad\text{or}\quad \Phi\colon A\times V \to V</math>
with two properties.  

<ol>
  <li>For any <math>g</math> in <math>G</math> (or <math>a</math> in <math>A</math>), the map <math display="block"> \begin{align}\Phi(g)\colon V& \to V\\
v & \mapsto \Phi(g, v)\end{align}</math>
is linear (over <math>\mathbb{F}</math>). 
  </li>
  <li>
  If we introduce the notation ''g'' · ''v'' for <math>\Phi</math> (''g'', ''v''), then for any ''g''<sub>1</sub>, ''g''<sub>2</sub> in ''G'' and ''v'' in ''V'':
<math display="block"> (2.1)\quad e \cdot v = v </math>
<math display="block"> (2.2)\quad g_1\cdot (g_2 \cdot v) = (g_1g_2) \cdot v </math>
where ''e'' is the [[identity element]] of ''G'' and ''g''<sub>1</sub>''g''<sub>2</sub> is the group product in ''G''.
</li>
</ol>
The definition for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (2.1) is omitted. Equation (2.2) 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 ''x''<sub>1</sub>, ''x''<sub>2</sub> in ''A'' and ''v'' in ''V'':
<math display="block"> (2.2')\quad x_1\cdot (x_2 \cdot v) - x_2\cdot (x_1 \cdot v) = [x_1,x_2] \cdot v </math>
where [''x''<sub>1</sub>, ''x''<sub>2</sub>] is the [[Lie algebra#Definition and first properties|Lie bracket]], which generalizes the matrix commutator ''MN'' &minus; ''NM''.

====Mapping====
The second way to define a representation focuses on the map ''φ'' sending ''g'' in ''G'' to a linear map ''φ''(''g''): ''V'' → ''V'', which satisfies
:<math> \varphi(g_1 g_2) = \varphi(g_1)\circ \varphi(g_2) \quad \text{for all }g_1,g_2 \in G </math>
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(''V'','''F''');<ref name=":1" />
* a representation of an associative algebra ''A'' on a vector space ''V'' is an [[algebra homomorphism]] ''φ'': ''A'' → End<sub>'''F'''</sub>(''V'');<ref name=":1" />
* a representation of a Lie algebra <math>\mathfrak{a}</math> on a vector space <math>V</math> is a [[Lie algebra homomorphism]] <math>\phi:\mathfrak{a}\to\mathfrak{gl}(V,\mathbb{F})</math>.

===Terminology===

The vector space ''V'' is called the '''representation space''' of ''φ'' and its [[dimension of a vector space|dimension]] (if finite) is called the '''dimension''' of the representation (sometimes ''degree'', as in <ref name="Serre 1977">{{Harvnb|Serre|1977}}.</ref>). 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 (linear algebra)|basis]] for ''V'' to identify ''V'' with '''F'''<sup>''n''</sup>, 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]].

=== Equivariant maps and isomorphisms ===
{{See also|Equivariant map}}

If ''V'' and ''W'' are vector spaces over '''F''', equipped with representations ''φ'' and ''ψ'' of a group ''G'', then an '''equivariant map''' from ''V'' to ''W'' is a linear map ''α'': ''V'' → ''W'' such that
:<math> \alpha( g\cdot v ) = g \cdot \alpha(v)</math>
for all ''g'' in ''G'' and ''v'' in ''V''. In terms of ''φ'': ''G'' → GL(''V'') and ''ψ'': ''G'' → GL(''W''), this means
:<math> \alpha\circ \varphi(g) = \psi(g)\circ \alpha </math>
for all ''g'' in ''G'', that is, the following [[Commutative diagram|diagram commutes]]:

:[[Image:Equivariant map commutative diagram.png|200px]]

Equivariant maps for representations of an associative or Lie algebra are defined similarly. If ''α'' is invertible, then it is said to be an [[isomorphism]], in which case ''V'' and ''W'' (or, more precisely, ''φ'' and ''ψ'') 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 {{mvar|G}}, it is on occasion called a {{math|''G''}}-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===
{{See also|Irreducible representation|simple module}}

If <math>(V,\psi)</math> is a representation of (say) a group <math>G</math>, and <math>W</math> is a linear subspace of <math>V</math> that is preserved by the action of <math>G</math> in the sense that for all <math>w \in W</math> and <math>g\in G</math>, <math>g\cdot w \in W</math> ([[Jean-Pierre Serre|Serre]] calls these <math>W</math> ''stable under'' <math>G</math><ref name="Serre 1977"/>), then <math>W</math> is called a ''[[subrepresentation]]'': by defining <math display="block">\phi:G \to \text{Aut}(W)</math> where <math>\phi(g)</math> is the restriction of <math>\psi(g)</math> to <math>W</math>, <math>(W,\phi)</math> is a representation of <math>G</math> and the inclusion of <math>W \hookrightarrow V</math> is an equivariant map. The [[quotient space (linear algebra)|quotient space]] <math>V/W</math> can also be made into a representation of <math>G</math>. If <math>V</math> has exactly two subrepresentations, namely the [[zero vector space|trivial subspace]] {0} and <math>V</math> itself, then the representation is said to be '''''irreducible'''''; if <math>V</math> has a proper nontrivial subrepresentation, the representation is said to be '''''reducible'''''.<ref>The representation {0} of dimension zero is considered to be neither reducible nor irreducible, just like the number 1 is considered to be neither composite nor [[prime number|prime]].</ref>

The definition of an irreducible representation implies [[Schur's lemma]]: an equivariant map <math display="block">\alpha: (V,\psi) \to (V',\psi')</math> between irreducible representations is either the [[zero map]] or an isomorphism, since its [[kernel (linear algebra)|kernel]] and [[image (mathematics)|image]] are subrepresentations. In particular, when <math>V = V'</math>, this shows that the equivariant [[endomorphism]]s of <math>V</math> 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 <math>V</math> is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if <math>V</math> 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 <math>(\mathbb{R},+)</math> has a two dimensional representation
<math display="block">\phi(a) = \begin{bmatrix}
1 & a \\
0 & 1
\end{bmatrix}</math>
This group has the vector <math>\begin{bmatrix} 1 & 0 \end{bmatrix}^\mathsf{T}</math> fixed by this homomorphism, but the complement subspace maps to
<math display="block">\begin{bmatrix} 0 \\ 1 \end{bmatrix} \mapsto \begin{bmatrix} a \\ 1 \end{bmatrix}</math>
giving only one irreducible subrepresentation. This is true for all [[Unipotent group|unipotent groups]].<ref>{{Cite book| last=Humphreys | first = James E.|url=https://www.worldcat.org/oclc/853255426|title=Linear Algebraic Groups|date=1975| publisher=Springer New York|isbn=978-1-4684-9443-3|location=New York, NY|oclc=853255426}}</ref>{{rp|112}}

=== Direct sums and indecomposable representations ===
{{See also|Direct sum|indecomposable module|semisimple module}}

If (''V'',''φ'') and (''W'',''ψ'') are representations of (say) a group ''G'', then the [[direct sum of vector spaces|direct sum]] of ''V'' and ''W'' is a representation, in a canonical way, via the equation
:<math> g\cdot (v,w) = (g\cdot v, g\cdot w).</math>

The [[direct sum of representations|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 "[[Weyl's theorem on complete reducibility|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 [[group extension|extensions]] of quotients by subrepresentations.

===Tensor products of representations===
{{main|Tensor product of representations}}
Suppose <math>\phi_1:G\rightarrow \mathrm{GL}(V_1)</math> and <math>\phi_2:G\rightarrow \mathrm{GL}(V_2)</math> are representations of a group <math>G</math>. Then we can form a representation <math>\phi_1\otimes\phi_2</math> of G acting on the [[tensor product]] vector space <math>V_1\otimes V_2</math> as follows:<ref>{{harvnb|Hall|2015}} Section 4.3.2</ref>
:<math>(\phi_1\otimes\phi_2)(g)=\phi_1(g)\otimes\phi_2(g)</math>.
If <math>\phi_1</math> and <math>\phi_2</math> are representations of a Lie algebra, then the correct formula to use is<ref>{{harvnb|Hall|2015}} Proposition 4.18 and Definition 4.19</ref>
:<math>(\phi_1\otimes\phi_2)(X)=\phi_1(X)\otimes I+I\otimes\phi_2(X)</math>.

This product can be recognized as the [[Tensor algebra#Coproduct|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_coefficients|Clebsch–Gordan theory]].

In the case of the [[Representation theory of SU(2)|representation theory of the group SU(2)]] (or equivalently, of its complexified Lie algebra <math>\mathrm{sl}(2;\mathbb{C})</math>), the decomposition is easy to work out.<ref>{{harvnb|Hall|2015}} Appendix C</ref> The irreducible representations are labeled by a parameter <math>l</math> that is a non-negative integer or half integer; the representation then has dimension <math>2l+1</math>. Suppose we take the tensor product of the representation of two representations, with labels <math>l_1</math> and <math>l_2,</math> where we assume <math>l_1\geq l_2</math>. Then the tensor product decomposes as a direct sum of one copy of each representation with label <math>l</math>, where <math>l</math> ranges from <math>l_1-l_2</math> to <math>l_1+l_2</math> in increments of 1. If, for example, <math>l_1=l_2=1</math>, then the values of <math>l</math> that occur are 0, 1, and 2. Thus, the tensor product representation of dimension <math>(2l_1+1) \times (2l_2+1) = 3\times 3=9</math> decomposes as a direct sum of a 1-dimensional representation <math>(l=0),</math> a 3-dimensional representation <math>(l=1),</math> and a 5-dimensional representation <math>(l=2)</math>.

==Branches and topics==
{{See also|Group representation}}

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:
# 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.
# Representation theory depends upon the nature of the vector space on which the algebraic object is represented. The most important distinction is between [[dimension (vector space)|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.
# Representation theory depends upon the type of [[field (mathematics)|field]] over which the vector space is defined. The most important cases are the field of complex numbers, the field of real numbers, [[finite field]]s, and fields of [[p-adic number]]s. Additional difficulties arise for fields of [[positive characteristic]] and for fields that are not [[algebraically closed]].

===Finite groups===
{{Main|Representation of a finite group}}

Group representations are a very important tool in the study of finite groups.<ref>{{Harvnb|Alperin|1986}}, {{Harvnb|Lam|1998}}, {{Harvnb|Serre|1977}}.</ref> They also arise in the applications of finite group theory to geometry and [[crystallographic group|crystallography]].<ref>{{Harvnb|Kim|1999}}.</ref> 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 (linear algebra)|projection]] ''π'' from ''W'' to ''V'' and replace it by its average ''π''<sub>''G''</sub> defined by
:<math> \pi_G(x) = \frac1{|G|}\sum_{g\in G} g\cdot \pi(g^{-1}\cdot x).</math>
''π''<sub>''G''</sub> 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''<sub>''φ''</sub>: ''G'' → '''F''' defined by
:<math>\chi_{\varphi}(g) = \mathrm{Tr}(\varphi(g))</math>
where <math>\mathrm{Tr}</math> is the [[trace of a matrix|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 field]]s, as long as the prime ''p'' is [[coprime]] to the [[group order|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]] <math>\langle\cdot,\cdot\rangle</math> on ''V'' in the sense that
:<math>\langle g\cdot v,g\cdot w\rangle = \langle v,w\rangle</math>
for all ''g'' in ''G'' and ''v'', ''w'' in ''W''. Hence any ''G''-representation is [[unitary representation|unitary]].

Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the [[orthogonal complement]] of a subrepresentation. When studying representations of groups that are not finite, the unitary representations provide a good generalization of the real and complex representations of a finite group.

Results such as Maschke's theorem and the unitary property that rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined. This can be done for [[compact group|compact topological groups]] (including compact Lie groups), using [[Haar measure]], and the resulting theory is known as [[abstract harmonic analysis]].

Over arbitrary fields, another class of finite groups that have a good representation theory are the [[finite groups of Lie type]]. Important examples are [[linear algebraic group]]s over finite fields. The representation theory of linear algebraic groups and [[Lie group]]s extends these examples to infinite-dimensional groups, the latter being intimately related to [[Lie algebra representation]]s. The importance of character theory for finite groups has an analogue in the theory of [[weight (representation theory)|weights]] for representations of Lie groups and Lie algebras.

Representations of a finite group ''G'' are also linked directly to algebra representations via the [[group ring|group algebra]] '''F'''[''G''], which is a vector space over '''F''' with the elements of ''G'' as a basis, equipped with the multiplication operation defined by the group operation, linearity, and the requirement that the group operation and scalar multiplication commute.

===Modular representations===
{{Main|Modular representation theory}}

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).<ref>{{Harvnb|Serre|1977|loc=Part III}}.</ref> 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 subgroup|Sylow 2-subgroup]]s were "too small".<ref>{{Harvnb|Alperin|1986}}.</ref>

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]].

===Unitary representations===
{{Main|Unitary representation}}

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]],<ref>See {{Harvnb|Weyl|1928}}.</ref> and this has inspired the development of the theory, most notably through the analysis of [[representation theory of the Poincaré group|representations of the Poincaré group]] by [[Eugene Wigner]].<ref>{{Harvnb|Wigner|1939}}.</ref> 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.<ref>{{Harvnb|Borel|2001}}.</ref>

A major goal is to describe the "[[unitary dual]]", the space of irreducible unitary representations of ''G''.<ref name=Knapp>{{Harvnb|Knapp|2001}}.</ref> The theory is most well-developed in the case that ''G'' is a [[locally compact]] (Hausdorff) [[topological group]] and the representations are [[strongly continuous]].<ref name=Folland/> For ''G'' abelian, the unitary dual is just the space of [[character theory|characters]], while for ''G'' compact, the [[Peter–Weyl theorem]] shows that the irreducible unitary representations are finite-dimensional and the unitary dual is discrete.<ref name="Peter–Weyl">{{Harvnb|Peter|Weyl|1927}}.</ref> For example, if ''G'' is the circle group ''S''<sup>1</sup>, 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 module]]s) 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 [[Semisimple Lie group|reductive]] [[Lie group]]s (discussed below), remains an important open problem in representation theory. It has been solved for many particular groups, such as [[representation theory of SL2(R)|SL(2,'''R''')]] and the [[representation theory of the Lorentz group|Lorentz group]].<ref>{{Harvnb|Bargmann|1947}}.</ref>

===Harmonic analysis===
{{Main|Abstract harmonic analysis}}

The duality between the circle group ''S''<sup>1</sup> and the integers '''Z''', or more generally, between a torus ''T''<sup>''n''</sup> and '''Z'''<sup>''n''</sup> 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 [[mathematical analysis|analysis]] of functions on [[local compactness|locally compact topological groups]] and related spaces.<ref name=Folland/>

A major goal is to provide a general form of the Fourier transform and the [[Plancherel theorem]]. This is done by constructing a [[measure (mathematics)|measure]] on the [[unitary dual]] and an isomorphism between the [[regular representation]] of ''G'' on the space L<sup>2</sup>(''G'') of [[square integrable]] functions on ''G'' and its representation on the [[L2-space|space of L<sup>2</sup> functions]] on the unitary dual. [[Pontrjagin duality]] and the [[Peter–Weyl theorem]] achieve this for abelian and compact ''G'' respectively.<ref name="Peter–Weyl"/><ref>{{Harvnb|Pontrjagin|1934}}.</ref>

Another approach involves considering all unitary representations, not just the irreducible ones. These form a [[category (mathematics)|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 group]]s and [[tannakian category|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 space]]s and provides a theory of [[automorphic form]]s (discussed below).

===Lie groups===
{{main|Representation of a Lie group}}
{{Lie groups}}

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.<ref name=Weyl>{{Harvnb|Weyl|1946}}.</ref> 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.<ref name="Sternberg"/>

The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply.<ref name=Knapp/> This theory can be extended to finite-dimensional representations of [[semisimple Lie group]]s using [[Weyl's unitary trick]]: each semisimple real Lie group ''G'' has a complexification, which is a complex Lie group ''G''<sup>c</sup>, 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]]).<ref name="Fulton-Harris">{{Harvnb|Fulton|Harris|1991}}.</ref> 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.

===Lie algebras===
{{Main|Lie algebra representation}}

A [[Lie algebra]] over a field '''F''' is a vector space over '''F''' equipped with a [[skew-symmetric graph|skew-symmetric]] [[bilinear operation]] called the [[Lie bracket]], which satisfies the [[Jacobi identity]]. Lie algebras arise in particular as [[tangent space]]s to [[Lie group]]s at the [[identity element]], leading to their interpretation as "infinitesimal symmetries".<ref name="Fulton-Harris"/> An important approach to the representation theory of Lie groups is to study the corresponding representation theory of Lie algebras, but representations of Lie algebras also have an intrinsic interest.<ref>{{Harvnb|Humphreys|1972a}}.</ref>

Lie algebras, like Lie groups, have a Levi decomposition into semisimple and solvable parts, with the representation theory of solvable Lie algebras being intractable in general. In contrast, the finite-dimensional representations of semisimple Lie algebras are completely understood, after work of [[Élie Cartan]]. A representation of a semisimple Lie algebra 𝖌 is analysed by choosing a [[Cartan subalgebra]], which is essentially a generic maximal subalgebra 𝖍 of 𝖌 on which the Lie bracket is zero ("abelian"). The representation of 𝖌 can be decomposed into [[weight (representation theory)|weight spaces]] that are [[eigenspace]]s for the action of 𝖍 and the infinitesimal analogue of characters. The structure of semisimple Lie algebras then reduces the analysis of representations to easily understood combinatorics of the possible weights that can occur.<ref name="Fulton-Harris"/>

====Infinite-dimensional Lie algebras====
{{See also|Affine Lie algebra|Kac–Moody algebra}}

There are many classes of infinite-dimensional Lie algebras whose representations have been studied. Among these, an important class are the Kac–Moody algebras.<ref>{{Harvnb|Kac|1990}}.</ref> They are named after [[Victor Kac]] and [[Robert Moody]], who independently discovered them. These algebras form a generalization of finite-dimensional [[semisimple Lie algebra]]s, and share many of their combinatorial properties. This means that they have a class of representations that can be understood in the same way as representations of semisimple Lie algebras.

Affine Lie algebras are a special case of Kac–Moody algebras, which have particular importance in mathematics and [[theoretical physics]], especially [[conformal field theory]] and the theory of [[exactly solvable model]]s. Kac discovered an elegant proof of certain combinatorial identities, [[Macdonald identities]], which is based on the representation theory of affine Kac–Moody algebras.

====Lie superalgebras====
{{Main|Representation of a Lie superalgebra}}

[[Lie superalgebra]]s are generalizations of Lie algebras in which the underlying vector space has a '''Z'''<sub>2</sub>-grading, and skew-symmetry and Jacobi identity properties of the Lie bracket are modified by signs. Their representation theory is similar to the representation theory of Lie algebras.<ref>{{Harvnb|Kac|1977}}.</ref>

===Linear algebraic groups===
{{See also|Linear algebraic group}}

Linear algebraic groups (or more generally, affine [[group scheme]]s) are analogues in algebraic geometry of [[Lie group]]s, but over more general fields than just '''R''' or '''C'''. In particular, over finite fields, they give rise to [[finite groups of Lie type]]. Although linear algebraic groups have a classification that is very similar to that of Lie groups, their representation theory is rather different (and much less well understood) and requires different techniques, since the [[Zariski topology]] is relatively weak, and techniques from analysis are no longer available.<ref>{{Harvnb|Humphreys|1972b}}, {{Harvnb|Jantzen|2003}}.</ref>

===Invariant theory===
{{Main|Invariant theory}}

Invariant theory studies [[Group action (mathematics)|actions]] on [[algebraic variety|algebraic varieties]] from the point of view of their effect on functions, which form representations of the group. Classically, the theory dealt with the question of explicit description of [[polynomial function]]s that do not change, or are ''invariant'', under the transformations from a given [[linear group]]. The modern approach analyses the decomposition of these representations into irreducibles.<ref>{{Harvnb|Olver|1999}}.</ref>

Invariant theory of [[infinite group]]s is inextricably linked with the development of [[linear algebra]], especially, the theories of [[quadratic form]]s and [[determinant]]s. Another subject with strong mutual influence is [[projective geometry]], where invariant theory can be used to organize the subject, and during the 1960s, new life was breathed into the subject by [[David Mumford]] in the form of his [[geometric invariant theory]].<ref>{{Harvnb|Mumford|Fogarty|Kirwan|1994}}.</ref>

The representation theory of [[semisimple Lie group]]s has its roots in invariant theory<ref name="Weyl" /> and the strong links between representation theory and algebraic geometry have many parallels in differential geometry, beginning with [[Felix Klein]]'s [[Erlangen program]] and [[Élie Cartan]]'s [[Cartan connection|connections]], which place groups and symmetry at the heart of geometry.<ref>{{Harvnb|Sharpe|1997}}.</ref> Modern developments link representation theory and invariant theory to areas as diverse as [[holonomy]], [[differential operator]]s and the theory of [[several complex variables]].

===Automorphic forms and number theory===
{{Main|Automorphic form}}

Automorphic forms are a generalization of [[modular form]]s to more general [[analytic function]]s, perhaps of [[several complex variables]], with similar transformation properties.<ref>{{Harvnb|Borel|Casselman|1979}}.</ref> The generalization involves replacing the modular group [[PSL2(R)|PSL<sub>2</sub> ('''R''')]] and a chosen [[congruence subgroup]] by a semisimple Lie group ''G'' and a [[discrete subgroup]] {{math|Γ}}. Just as modular forms can be viewed as [[differential form]]s on a quotient of the [[upper half space]] '''''H''''' = PSL<sub>2</sub> ('''R''')/SO(2), automorphic forms can be viewed as differential forms (or similar objects) on {{math|Γ\''G''/''K''}}, where ''K'' is (typically) a [[maximal compact subgroup]] of ''G''. Some care is required, however, as the quotient typically has singularities. The quotient of a semisimple Lie group by a compact subgroup is a [[symmetric space]] and so the theory of automorphic forms is intimately related to harmonic analysis on symmetric spaces.

Before the development of the general theory, many important special cases were worked out in detail, including the [[Hilbert modular form]]s and [[Siegel modular form]]s. Important results in the 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. The subsequent notion of "automorphic representation" has proved of great technical value for dealing with the case that ''G'' is an [[algebraic group]], treated as an [[adelic algebraic group]]. As a result, an entire philosophy, the [[Langlands program]] has developed around the relation between representation and number theoretic properties of automorphic forms.<ref>{{Harvnb|Gelbart|1984}}.</ref>

===Associative algebras===
{{Main|Algebra representation}}

In one sense, [[associative algebra]] representations generalize both representations of groups and Lie algebras. A representation of a group induces a representation of a corresponding [[group ring]] or [[group ring|group algebra]], while representations of a Lie algebra correspond bijectively to representations of its [[universal enveloping algebra]]. However, the representation theory of general associative algebras does not have all of the nice properties of the representation theory of groups and Lie algebras.

====Module theory====
{{Main|Module theory}}

When considering representations of an associative algebra, one can forget the underlying field, and simply regard the associative algebra as a ring, and its representations as modules. This approach is surprisingly fruitful: many results in representation theory can be interpreted as special cases of results about modules over a ring.

====Hopf algebras and quantum groups====
{{Main|Representation theory of Hopf algebras}}

[[Hopf algebra]]s provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and Lie algebras as special cases. In particular, the tensor product of two representations is a representation, as is the dual vector space.

The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopf algebras are known as [[quantum group]]s, although this term is often restricted to certain Hopf algebras arising as deformations of groups or their universal enveloping algebras. The representation theory of quantum groups has added surprising insights to the representation theory of Lie groups and Lie algebras, for instance through the [[crystal basis]] of Kashiwara.

==History==

{{main|History of representation theory}}
<!-- To do: expand the main article, and summarize it here -->

==Generalizations==

===Set-theoretic representations===
{{Main|Group action (mathematics)}}

A ''set-theoretic representation'' (also known as a [[Group action (mathematics)|group action]] or ''permutation representation'') of a [[group (mathematics)|group]] ''G'' on a [[set (mathematics)|set]] ''X'' is given by a [[function (mathematics)|function]] ''ρ'' from ''G'' to ''X''<sup>''X''</sup>, the [[set (mathematics)|set]] of [[function (mathematics)|function]]s from ''X'' to ''X'', such that for all ''g''<sub>1</sub>, ''g''<sub>2</sub> in ''G'' and all ''x'' in ''X'':

:<math>\rho(1)[x] = x</math>
:<math>\rho(g_1 g_2)[x]=\rho(g_1)[\rho(g_2)[x]].</math>

This condition and the axioms for a group imply that ''ρ''(''g'') is a [[bijection]] (or [[permutation]]) for all ''g'' in ''G''. Thus we may equivalently define a permutation representation to be a [[group homomorphism]] from G to the [[symmetric group]] S<sub>''X''</sub> of ''X''.

===Representations in other categories===
{{See also|Category theory}}

Every group ''G'' can be viewed as a [[category (mathematics)|category]] with a single object; [[morphism]]s in this category are just the elements of ''G''. Given an arbitrary category ''C'', a ''representation'' of ''G'' in ''C'' is a [[functor]] from ''G'' to ''C''. Such a functor selects an object ''X'' in ''C'' and a group homomorphism from ''G'' to Aut(''X''), the [[automorphism group]] of ''X''.

In the case where ''C'' is '''Vect'''<sub>'''F'''</sub>, the [[category of vector spaces]] over a field '''F''', this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of ''G'' in the [[category of sets]].

For another example consider the [[category of topological spaces]], '''Top'''. Representations in '''Top''' are homomorphisms from ''G'' to the [[homeomorphism]] group of a topological space ''X''.

Three types of representations closely related to linear representations are:
*[[projective representation]]s: in the category of [[projective space]]s. These can be described as "linear representations [[up to]] scalar transformations".
*[[affine representation]]s: in the category of [[affine space]]s. For example, the [[Euclidean group]] acts affinely upon [[Euclidean space]].
*[[corepresentations of unitary and antiunitary groups]]: in the category of complex vector spaces with morphisms being linear or [[antilinear]] transformations.

===Representations of categories===

{{See also|Quiver (mathematics)}}

Since groups are categories, one can also consider representation of other categories. The simplest generalization is to [[monoid]]s, which are categories with one object. Groups are monoids for which every morphism is invertible. General monoids have representations in any category. In the category of sets, these are [[monoid action]]s, but monoid representations on vector spaces and other objects can be studied.

More generally, one can relax the assumption that the category being represented has only one object. In full generality, this is simply the theory of [[functor]]s between categories, and little can be said.

One special case has had a significant impact on representation theory, namely the representation theory of quivers.<ref name=SSA/> A quiver is simply a [[directed graph]] (with loops and multiple arrows allowed), but it can be made into a category (and also an algebra) by considering paths in the graph. Representations of such categories/algebras have illuminated several aspects of representation theory, for instance by allowing non-semisimple representation theory questions about a group to be reduced in some cases to semisimple representation theory questions about a quiver.

==See also==
{{Div col|colwidth=20em}}
* [[Galois representation]]
* [[Glossary of representation theory]]
* [[Group representation]]
* [[Itô's theorem]]
* [[List of representation theory topics]]
* [[List of harmonic analysis topics]]
* [[Numerical analysis]]
* [[Philosophy of cusp forms]]
* [[Representation (mathematics)]]
* [[Representation theorem]]
* [[Universal algebra]]
{{Div col end}}

==Notes==

{{Reflist|2}}

==References==

* {{Citation|title= Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups|first=J. L.|last= Alperin|author-link=J. L. Alperin|publisher=Cambridge University Press|year= 1986|isbn=978-0-521-44926-7}}.
* {{Citation|first=V.|last=Bargmann|title=Irreducible unitary representations of the Lorenz group|journal= [[Annals of Mathematics]]|volume=48|year=1947|pages=568–640|doi=10.2307/1969129|issue=3|jstor=1969129}}.
* {{Citation|title=Essays in the History of Lie Groups and Algebraic Groups|first=Armand|last= Borel|author-link=Armand Borel|publisher=American Mathematical Society|year= 2001|isbn=978-0-8218-0288-5}}.
* {{Citation|title=Automorphic Forms, Representations, and L-functions|first1=Armand |last1=Borel|first2= W.|last2=Casselman|publisher=American Mathematical Society|year= 1979|isbn=978-0-8218-1435-2}}.
* {{Citation|title=Representation Theory of Finite Groups and Associative Algebras|first1=Charles W.|last1=Curtis|author1-link=Charles W. Curtis|first2=Irving|last2=Reiner|author2-link=Irving Reiner|publisher=John Wiley & Sons (Reedition 2006 by AMS Bookstore)|year=1962|isbn=978-0-470-18975-7|url-access=registration|url=https://archive.org/details/representationth11curt}}.<!---citation template accepts only one isbn---(ISBN 978-0821840665)--->
* {{Citation|title=A Course in Abstract Harmonic Analysis|first=Gerald B.|last= Folland|publisher=CRC Press|year= 1995|isbn=978-0-8493-8490-5}}.
* {{Fulton-Harris}}.
* {{Citation|first=Stephen|last=Gelbart|author-link= Stephen Gelbart |title=An Elementary Introduction to the Langlands Program|journal=Bulletin of the American Mathematical Society|volume=10|issue=2|year=1984|pages=177–219|doi=10.1090/S0273-0979-1984-15237-6|doi-access=free}}.
* {{Citation|last2=Wallach|first2=Nolan R.|last1=Goodman|first1=Roe|year=1998|title=Representations and Invariants of the Classical Groups|publisher= Cambridge University Press|isbn= 978-0-521-66348-9}}.
* {{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn= 978-3319134666}}
* {{Citation|first=Sigurdur|last=Helgason|title=Differential Geometry, Lie groups and Symmetric Spaces|publisher=Academic Press|year=1978|isbn=978-0-12-338460-7}}
* {{citation|title=Introduction to Lie Algebras and Representation Theory|first=James E.|last=Humphreys|publisher=Birkhäuser|year=1972a|isbn=978-0-387-90053-7|url-access=registration|url=https://archive.org/details/introductiontoli00jame}}.
* {{citation | last1=Humphreys | first1=James E. | title=Linear Algebraic Groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90108-4 |mr=0396773 | year=1972b | volume=21}}
* {{citation |last1=James|first1=Gordon|author-link2=Martin Liebeck |last2=Liebeck|first2= Martin | title=Representations and Characters of Groups | location=Cambridge | publisher=Cambridge University Press | year=1993 | isbn=978-0-521-44590-0}}.
* {{Citation|title=Representations of Algebraic Groups|first=Jens Carsten|last= Jantzen|publisher=American Mathematical Society|year= 2003|isbn=978-0-8218-3527-2}}.
* {{Citation|last=Kac|first= Victor G.|author-link=Victor Kac|title= Lie superalgebras|journal= [[Advances in Mathematics]]|volume= 26 |year=1977|issue= 1|pages=8&ndash;96|doi=10.1016/0001-8708(77)90017-2|doi-access=free}}.
* {{Citation|last=Kac|first= Victor G.|title=Infinite Dimensional Lie Algebras|edition=3rd|publisher=Cambridge University Press|year= 1990|isbn=978-0-521-46693-6}}.
* {{Citation|title=Group Theoretical Methods and Applications to Molecules and Crystals: And Applications to Molecules and Crystals|first=Shoon Kyung|last=Kim|publisher=Cambridge University Press|year= 1999|isbn=978-0-521-64062-6}}.
* {{Citation|title=Representation Theory of Semisimple Groups: An Overview Based on Examples|first=Anthony W.|last= Knapp|author-link=Anthony W. Knapp|publisher= Princeton University Press|year= 2001|isbn=978-0-691-09089-4}}.
* {{Citation|title=Linear Algebra and Geometry|first1=A. I.|last1=Kostrikin|author-link=Alexei Kostrikin|first2=Yuri I.|last2= Manin|author2-link=Yuri Manin|publisher=Taylor & Francis|year= 1997|isbn=978-90-5699-049-7}}.
* {{Citation|title=Representations of finite groups: a hundred years|first=T. Y.|last=Lam|journal=Notices of the AMS|volume = 45| issue= 3,4|year=1998|pages =[https://www.ams.org/notices/199803/lam.pdf 361&ndash;372 (Part I)], [https://www.ams.org/notices/199804/lam2.pdf 465&ndash;474 (Part II)]}}.
* {{cite book | last=Lyubich | first=Yurii I. | title=Introduction to the Theory of Banach Representations of Groups |series=Operator Theory: Advances and Applications|volume=30| publisher=Birkhauser | publication-place=Basel | date=1988 | isbn=978-3-7643-2207-6}}
*{{citation | last1=Mumford | first1=David | author1-link=David Mumford | last2=Fogarty | first2=J. | last3=Kirwan | first3=F. | title=Geometric invariant theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)] | isbn=978-3-540-56963-3 |mr=0214602<!--(1st ed. 1965)--> | year=1994 | volume=34}}; {{MathSciNet|id=0719371}} (2nd ed.); {{MathSciNet | id = 1304906}}(3rd ed.)
* {{citation | last=Olver|first= Peter J. |author-link=Peter J. Olver | title=Classical invariant theory | location=Cambridge | publisher=Cambridge University Press | year=1999 | isbn= 978-0-521-55821-1}}.
* {{Citation|first1=F.|last1=Peter|first2=Hermann|last2=Weyl|title=Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe|journal=[[Mathematische Annalen]]|volume=97|year=1927|issue=1|pages=737–755|doi=10.1007/BF01447892|doi-access=|s2cid=120013521}}.
* {{Citation|first=Lev S.|last= Pontrjagin|author-link=Lev Pontryagin|title=The theory of topological commutative groups|journal=[[Annals of Mathematics]]|volume=35|year= 1934|pages= 361–388|doi=10.2307/1968438|issue=2|jstor=1968438}}.
* {{Citation|title=Representation Theory and Harmonic Analysis on Semisimple Lie Groups|first1=Paul|last1= Sally|author1-link=Paul Sally|first2= David A.|last2= Vogan|author2-link=David Vogan|publisher=American Mathematical Society|year=1989|isbn=978-0-8218-1526-7}}.
* {{citation | author-link=Jean-Pierre Serre | first=Jean-Pierre | last=Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn=978-0387901909 | url-access=registration | url=https://archive.org/details/linearrepresenta1977serr }}.
* {{Citation| title= Differential Geometry: Cartan's Generalization of Klein's Erlangen Program|first=Richard W.|last= Sharpe|publisher=Springer|year= 1997|isbn= 978-0-387-94732-7}}.
* {{citation | title=Elements of the Representation Theory of Associative Algebras|first1=Daniel|last1= Simson|first2= Andrzej|last2=Skowronski|first3= Ibrahim|last3= Assem|publisher= Cambridge University Press|year= 2007|isbn=978-0-521-88218-7}}.
* {{citation|title=Group Theory and Physics|first=Shlomo|last=Sternberg|author-link=Shlomo Sternberg|publisher=Cambridge University Press|year=1994|isbn=978-0-521-55885-3|url-access=registration|url=https://archive.org/details/grouptheoryphysi0000ster}}.
* {{cite book|title=Group Theory in Physics|edition=1st|location=New Jersey·London·Singapore·Hong Kong|year=1985|isbn=978-9971966577|publisher=[[World Scientific Publishing|World Scientific]]|last=Tung|first=Wu-Ki}}
* {{citation | title= Gruppentheorie und Quantenmechanik |title-link=Gruppentheorie und Quantenmechanik |first=Hermann|last= Weyl|author-link=Hermann Weyl|edition=The Theory of Groups and Quantum Mechanics, translated H.P. Robertson, 1931|publisher= S. Hirzel, Leipzig (reprinted 1950, Dover)|year=1928|isbn=978-0-486-60269-1}}.
* {{Citation|title=The Classical Groups: Their Invariants and Representations|first=Hermann|last= Weyl|year=1946|edition=2nd|publisher = Princeton University Press (reprinted 1997)| isbn= 978-0-691-05756-9}}.
* {{Citation|first= Eugene P.|last=Wigner|author-link=Eugene Wigner|title=On unitary representations of the inhomogeneous Lorentz group|journal=[[Annals of Mathematics]]| volume=40|pages=149–204|year=1939|doi=10.2307/1968551|issue=1|jstor= 1968551|bibcode=1939AnMat..40..149W |s2cid=121773411 }}.

==External links==
{{Sister project links| wikt=no | commons=no | b=no | n=no | q=Representation theory | s=no | v=no | voy=no | species=no | d=no}}

* {{springer|title=Representation theory|id=p/r081480}}
* [[Alexander Kirillov Jr.]], [http://www.math.sunysb.edu/~kirillov/liegroups/ An introduction to Lie groups and Lie algebras] (2008).  Textbook, preliminary version pdf downloadable from author's home page.
* Kevin Hartnett, [https://www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609/ (2020), article on representation theory in Quanta magazine]

{{Analysis-footer}}
{{authority control}}

[[Category:Representation theory| ]]

{{bots|deny=Yobot}}