Representation Theory

Representation theory is a branch of mathematics that studies algebraic structures by realizing their elements as linear maps on vector spaces. It is especially concerned with representations of groups, associative algebras, and Lie algebras. This approach translates questions about abstract operations and symmetry into problems in linear algebra, where matrices, invariant subspaces, and decomposition techniques provide concrete tools. Central questions include how representations can be classified, how they decompose into simpler representations, and how different representations are related. (ocw.mit.edu)

Definition and basic examples

For a group $G$ and a vector space $V$ over a field $k$, a linear representation is a homomorphism

where $\operatorname{GL}(V)$ is the group of invertible linear maps from $V$ to itself. Thus

Equivalently, a representation is a group action on $V$ by linear transformations. Choosing a basis in a finite-dimensional space expresses these transformations as invertible matrices. A representation is faithful when $\rho$ is injective: distinct group elements then act differently. Faithfulness is not required in the definition. (ocw.mit.edu)

The trivial representation sends every group element to the identity. A permutation representation arises when a group permutes a set: one takes basis vectors indexed by the set and permutes them accordingly. For a finite group, the regular representation has basis $\{e_g:g\in G\}$, with

Its dimension is $|G|$, and it contains every irreducible complex representation of $G$. (ocw.mit.edu)

For an associative unital $k$-algebra $A$, a representation is instead a unital algebra homomorphism

This is equivalent to making $V$ a left module over $A$. Group representations fit this framework through the group algebra $k[G]$, whose multiplication extends the group multiplication linearly. (ocw.mit.edu)

Equivalence, irreducibility, and decomposition

Two representations $(V,\rho)$ and $(W,\sigma)$ are equivalent if there is a linear isomorphism $T:V\to W$ satisfying

for every $g\in G$. Such a map is called an intertwiner. In matrix form, equivalent representations differ by a simultaneous change of basis, rather than by separate changes for individual group elements. (arxiv.org)

A linear subspace $U\subseteq V$ is invariant if $\rho(g)U\subseteq U$ for every $g$. A nonzero representation is irreducible, or simple, if its only invariant subspaces are $0$ and $V$. It is completely reducible, or semisimple, if it is a direct sum of irreducible representations. An indecomposable representation cannot be written as a direct sum of two nonzero subrepresentations; it need not be irreducible. (ocw.mit.edu)

Schur’s lemma states that a nonzero intertwiner between irreducible representations is an isomorphism. For a finite-dimensional irreducible representation over an algebraically closed field, every intertwining endomorphism is a scalar multiple of the identity. More generally, finite-dimensional representations have composition series with irreducible successive quotients. The Jordan–Hölder theorem makes these factors, with multiplicities, independent of the chosen series, but the factors alone need not determine the representation. (ocw.mit.edu)

Finite groups and character theory

For a finite group $G$, Maschke’s theorem guarantees complete reducibility when the characteristic of the field does not divide $|G|$. In particular, every finite-dimensional complex representation of a finite group decomposes into irreducibles. The proof averages a projection over the group to obtain an invariant complement; division by $|G|$ is the essential step. (math.ucla.edu)

Over the complex numbers, the character of a finite-dimensional representation is

using the trace. Characters are constant on conjugacy classes. The irreducible characters form an orthonormal basis for the complex-valued class functions under

Consequently, the multiplicity of an irreducible representation $W$ in $V$ is $\langle\chi_V,\chi_W\rangle$. If $d_1,\ldots,d_r$ are the dimensions of the inequivalent irreducible complex representations, then

The number $r$ equals the number of conjugacy classes. (ocw.mit.edu)

For example, the symmetric group $S_3$ has three irreducible complex representations: the one-dimensional trivial and sign representations, and a two-dimensional standard representation. The last is realized inside its permutation representation on $\mathbb C^3$ as

Their dimensions satisfy $1^2+1^2+2^2=6$. (ocw.mit.edu)

Constructing and relating representations

Representations can be combined by direct sums and tensor products. On a tensor product,

Their characters therefore satisfy

Determining the irreducible components of tensor products is a major recurring problem. (ocw.mit.edu)

If $H$ is a subgroup of $G$, restriction views a $G$-representation as an $H$-representation. Conversely, induction constructs a $G$-representation from an $H$-representation. Frobenius reciprocity relates the corresponding intertwiner spaces:

Another important relationship is Schur–Weyl duality: commuting actions of general linear and symmetric groups on tensor powers organize their decompositions together, linking representation theory with combinatorics. (ocw.mit.edu)

Lie groups, Lie algebras, and geometry

Representations of a Lie group incorporate its continuous structure; finite-dimensional representations are ordinarily required to be continuous. Differentiating a representation at the identity gives a representation of its Lie algebra, a linear map

that preserves brackets:

For a connected, simply connected Lie group, every finite-dimensional representation of its Lie algebra integrates uniquely to a group representation. For other groups, global topology can obstruct integration. (arxiv.org)

Finite-dimensional irreducible representations of complex semisimple Lie algebras are classified by dominant integral highest weights. Weights describe simultaneous eigenvalues for a chosen Cartan subalgebra, while root operators relate the corresponding weight spaces. This converts classification into structured algebraic and combinatorial data. For $\mathfrak{sl}_2(\mathbb C)$, there is exactly one irreducible representation of each positive integer dimension. (arxiv.org)

Geometric representation theory uses spaces with symmetry to construct and study representations. One example is the Borel–Weil construction, which realizes irreducible representations of complex reductive groups through sections of suitable line bundles on flag varieties. This connects representation theory with algebraic geometry, rather than treating representations solely as collections of matrices. (math.columbia.edu)

Modular representations and quivers

When the field has positive characteristic dividing a finite group’s order, complete reducibility can fail. Modular representation theory therefore studies not only simple representations but also how they can be joined through nonsplit extensions. This makes the distinction between irreducible and indecomposable representations essential. (math.ucla.edu)

A concrete example occurs for the cyclic group $C_p$ over a field of characteristic $p$. Its generator can act on $k^2$ by

Because $J^p=I$, this defines a representation. It has an invariant line but no invariant complementary line: it is reducible and indecomposable. (math.mit.edu)

A quiver is a directed graph, and a quiver representation assigns a vector space to every vertex and a linear map to every arrow. Quivers provide a framework for classifying systems of linear maps and representations of many finite-dimensional algebras. Gabriel’s theorem states that a finite quiver over an algebraically closed field has only finitely many isomorphism classes of indecomposable finite-dimensional representations precisely when its underlying graph is a disjoint union of Dynkin diagrams of types $A$, $D$, and $E$. (ocw.mit.edu)

Historical development and applications

The modern theory of finite-group representations emerged from Georg Frobenius’s work on group characters and group determinants in 1896–1897, prompted by questions from Richard Dedekind. Subsequent work by Issai Schur and Hermann Weyl developed its algebraic structure and its connections with Lie groups. The subject expanded from finite-group symmetry into a unified study of groups, algebras, and their linear actions. (math.mit.edu)

In quantum mechanics, representations describe how symmetry transformations act on state spaces; rotational representations organize angular momentum and spin. In spectroscopy and crystallography, representations organize states and modes according to symmetry. Representation theory also connects number theory, geometry, and combinatorics. These applications depend on more than assigning matrices: they use decomposition to separate a system into components with distinct transformation properties. (math.mit.edu)

References

1. Introduction to Representation Theoryocw.mit.edu
2. Chapter 1: Basic notions of representation theoryocw.mit.edu
3. Chapter 2: General results of representation theoryocw.mit.edu
4. Chapter 3: Representations of finite groups: basic resultsocw.mit.edu
5. Chapter 4: Representations of finite groups: further resultsocw.mit.edu
6. Chapter 5: Quiver Representationsocw.mit.edu
7. An Elementary Introduction to Groups and Representationsarxiv.org
8. Abstract Algebra — Chapter 13 Representation theorymath.ucla.edu
9. The Modular Representation Theory of Cyclic Groupsmath.mit.edu
10. Introduction to representation theorymath.mit.edu
11. Lie Groups and Representations: Mathematics G4344math.columbia.edu