Chapter 106. Representation Theory Basics

Representation theory studies algebraic objects by letting them act on vector spaces. Instead of studying a group, algebra, or Lie algebra only through its abstract definition, one represents its elements as linear transformations. This brings matrices, eigenvalues, invariant subspaces, decompositions, and linear algebra into the study of symmetry. Standard introductions describe representation theory as the study of groups or algebraic structures through their actions on vector spaces.

106.1 The Basic Idea

Let $G$ be a group.

A representation of $G$ assigns to each element $g\in G$ an invertible linear map on a vector space $V$ . The assignment must respect the group multiplication.

Thus a representation converts abstract multiplication in $G$ into composition of linear maps.

\rho : G \to GL(V)

is a representation, then

\rho(gh)=\rho(g)\rho(h)

for all $g,h\in G$ , and

\rho(e)=I_V.

Here $e$ is the identity element of $G$ , and $I_V$ is the identity map on $V$ .

The vector space $V$ is called the representation space.

106.2 Linear Actions

A representation may also be described as a linear action.

A group $G$ acts linearly on $V$ if each $g\in G$ sends vectors in $V$ to vectors in $V$ , and

g(v+w)=gv+gw,

g(cv)=c(gv),

e v=v,

and

g(hv)=(gh)v.

The notation $gv$ means the action of $g$ on the vector $v$ .

The representation map and the action notation are equivalent. If

\rho : G\to GL(V),

then

gv=\rho(g)v.

Linear actions are often called $G$ -modules.

106.3 Matrix Representations

If $V$ has finite dimension $n$ , then choosing a basis identifies

GL(V)

with

GL_n(F),

the group of invertible $n\times n$ matrices.

A representation can then be written as

\rho : G\to GL_n(F).

Each group element becomes a matrix.

The condition

\rho(gh)=\rho(g)\rho(h)

says that group multiplication is represented by matrix multiplication.

Thus representation theory studies abstract symmetry through concrete matrices.

106.4 First Example

Let

G=\mathbb{Z}/2\mathbb{Z}=\{0,1\}

with addition modulo $2$ .

Define a representation on $\mathbb{R}^2$ by

\rho(0)= \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix},

and

\rho(1)= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}.

The element $1$ acts as reflection across the $x$ -axis.

Since

1+1=0

in $\mathbb{Z}/2\mathbb{Z}$ , we need

\rho(1)^2=\rho(0).

Indeed,

\begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}^2 = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}.

Thus this is a representation.

106.5 Trivial Representation

Every group has a trivial representation on any vector space $V$ .

It is defined by

\rho(g)=I_V

for every $g\in G$ .

In this representation, every group element acts as the identity.

The trivial representation contains no visible motion. It is still important because it often appears as a component inside larger representations.

For example, invariant vectors form copies of the trivial representation.

106.6 Faithful Representations

A representation

\rho:G\to GL(V)

is faithful if $\rho$ is injective.

This means different group elements act by different linear transformations.

Equivalently,

\rho(g)=I_V

only when

g=e.

A faithful representation loses no group information. It realizes the group as a concrete group of linear transformations.

A non-faithful representation collapses some group elements together.

106.7 Kernel of a Representation

The kernel of a representation is

\ker \rho = \{g\in G:\rho(g)=I_V\}.

It consists of all group elements that act trivially on $V$ .

The kernel is a normal subgroup of $G$ .

A representation is faithful exactly when

\ker\rho=\{e\}.

Thus the kernel measures how much of the group the representation fails to see.

106.8 Subrepresentations

Let $V$ be a representation of $G$ .

A subspace

W\subseteq V

is a subrepresentation if it is invariant under the action of $G$ :

gW\subseteq W

for every $g\in G$ .

Equivalently, for every $w\in W$ ,

gw\in W.

If $W$ is invariant, then the action of $G$ restricts to $W$ , so $W$ becomes a representation in its own right.

Invariant subspaces are the representation-theoretic analogue of subspaces preserved by a linear operator.

106.9 Irreducible Representations

A nonzero representation $V$ is irreducible if its only subrepresentations are

0

and

V.

An irreducible representation cannot be decomposed into smaller invariant pieces.

Irreducible representations are the basic building blocks of representation theory.

This is analogous to prime numbers in arithmetic or simple modules in algebra.

A main problem in representation theory is to classify irreducible representations of a given algebraic object.

106.10 Direct Sums

If $V$ and $W$ are representations of $G$ , their direct sum

V\oplus W

is also a representation.

The action is defined by

g(v,w)=(gv,gw).

In matrix form, if $g$ acts on $V$ by $A_g$ and on $W$ by $B_g$ , then $g$ acts on $V\oplus W$ by the block diagonal matrix

\begin{bmatrix} A_g&0\\ 0&B_g \end{bmatrix}.

Direct sums combine representations without mixing their components.

106.11 Decomposable and Indecomposable Representations

A representation $V$ is decomposable if it can be written as a direct sum

V=W_1\oplus W_2

where $W_1$ and $W_2$ are nonzero subrepresentations.

If no such decomposition exists, $V$ is indecomposable.

Every irreducible representation is indecomposable.

The converse may fail. A representation may have nontrivial subrepresentations but still resist splitting as a direct sum.

This distinction becomes important over fields or algebras where complete reducibility fails.

106.12 Complete Reducibility

A representation is completely reducible if it is a direct sum of irreducible representations.

Thus

V\cong V_1\oplus\cdots\oplus V_r,

where each $V_i$ is irreducible.

For finite groups over fields of characteristic $0$ , every finite-dimensional representation is completely reducible. This is Maschke’s theorem.

Complete reducibility allows representation theory to reduce many questions to irreducible representations.

Without complete reducibility, extensions between representations become part of the theory.

106.13 Homomorphisms of Representations

Let $V$ and $W$ be representations of $G$ .

A homomorphism of representations is a linear map

T:V\to W

such that

T(gv)=gT(v)

for every $g\in G$ and every $v\in V$ .

Such maps are also called equivariant maps or intertwining maps.

They preserve both the vector-space structure and the group action.

In terms of representation maps,

T\rho_V(g)=\rho_W(g)T.

This equation says that $T$ commutes with the action of $G$ .

106.14 Isomorphic Representations

Two representations $V$ and $W$ are isomorphic if there exists an invertible representation homomorphism

T:V\to W.

That means

T(gv)=gT(v)

and $T$ is a vector-space isomorphism.

In matrix terms, isomorphic representations differ only by a change of basis.

\rho_W(g)=P^{-1}\rho_V(g)P

for every $g\in G$ , then the two representations are equivalent.

Thus representation theory classifies actions up to change of coordinates.

106.15 Schur’s Lemma

Schur’s lemma is one of the fundamental results of representation theory.

Let $V$ and $W$ be irreducible representations over an algebraically closed field.

T:V\to W

is a representation homomorphism, then either

T=0

or $T$ is an isomorphism.

In particular, if $V$ is irreducible, then every endomorphism

T:V\to V

that commutes with the group action is scalar multiplication:

T=\lambda I.

Schur’s lemma explains why irreducible representations behave like atomic objects.

106.16 Characters

For a finite-dimensional representation

\rho:G\to GL(V),

the character is the function

\chi_\rho:G\to F

defined by

\chi_\rho(g)=\operatorname{tr}(\rho(g)).

The character records the trace of each representing matrix.

Characters are useful because trace is unchanged under change of basis.

Thus isomorphic representations have the same character.

For many classes of finite groups, characters provide a powerful way to decompose and classify representations.

106.17 Example of a Character

Consider the representation of

\mathbb{Z}/2\mathbb{Z}

on $\mathbb{R}^2$ from Section 106.4.

We have

\rho(0)= \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}, \qquad \rho(1)= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}.

The character is

\chi(0)=\operatorname{tr}(\rho(0))=2,

and

\chi(1)=\operatorname{tr}(\rho(1))=0.

The value at the identity equals the dimension of the representation.

106.18 Representations of Lie Algebras

Representation theory also applies to Lie algebras.

If $\mathfrak g$ is a Lie algebra, a representation of $\mathfrak g$ on $V$ is a Lie algebra homomorphism

\rho:\mathfrak g\to\mathfrak{gl}(V).

This means

\rho([x,y]) = [\rho(x),\rho(y)]

for all $x,y\in\mathfrak g$ .

The bracket on the right is the commutator bracket:

[A,B]=AB-BA.

Thus Lie algebra elements act as linear operators whose commutators reproduce the Lie bracket. This is the standard definition for Lie algebra representations.

106.19 The Adjoint Representation

Every Lie algebra $\mathfrak g$ has a natural representation on itself.

Define

\operatorname{ad}_x:\mathfrak g\to\mathfrak g

\operatorname{ad}_x(y)=[x,y].

The map

\operatorname{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g)

is called the adjoint representation.

The Jacobi identity ensures that

\operatorname{ad}_{[x,y]} = [\operatorname{ad}_x,\operatorname{ad}_y].

Thus the adjoint representation is a genuine Lie algebra representation.

106.20 Representations of Associative Algebras

Let $A$ be an associative algebra over $F$ .

A representation of $A$ on a vector space $V$ is an algebra homomorphism

\rho:A\to \operatorname{End}_F(V).

Equivalently, $V$ is an $A$ -module.

This means elements of $A$ act as linear maps on $V$ , and multiplication in $A$ corresponds to composition of linear maps.

Group representations can often be studied through the group algebra $F[G]$ . Lie algebra representations can often be studied through the universal enveloping algebra. This connects group, Lie, and associative algebra representation theories.

106.21 Invariant Vectors

Let $V$ be a representation of $G$ .

A vector $v\in V$ is invariant if

gv=v

for every $g\in G$ .

The set of invariant vectors is

V^G = \{v\in V:gv=v\text{ for all }g\in G\}.

This is a subspace of $V$ .

Invariant vectors form the part of the representation on which the group acts trivially.

In applications, invariant vectors often correspond to conserved quantities, symmetric tensors, or fixed features.

106.22 Dual Representations

If $V$ is a representation of $G$ , then the dual space

V^*

also carries a representation.

For $f\in V^*$ , define

(gf)(v)=f(g^{-1}v).

The inverse is needed so that the group action law holds.

The dual representation describes how linear functionals transform when vectors transform.

Dual representations are essential in tensor algebra, invariant theory, and differential geometry.

106.23 Tensor Product Representations

If $V$ and $W$ are representations of $G$ , then

V\otimes W

is also a representation.

The action is defined on pure tensors by

g(v\otimes w)=(gv)\otimes(gw).

This extends linearly to all of $V\otimes W$ .

Tensor products allow one to build new representations from old ones.

They also lead to decomposition problems: given $V\otimes W$ , determine its irreducible components.

106.24 Symmetric and Exterior Powers

If $V$ is a representation of $G$ , then the symmetric powers

S^k(V)

and exterior powers

\Lambda^k(V)

are also representations.

The group action is induced by

g(v_1\cdots v_k) = (gv_1)\cdots(gv_k)

on symmetric powers and

g(v_1\wedge\cdots\wedge v_k) = (gv_1)\wedge\cdots\wedge(gv_k)

on exterior powers.

These constructions are central in geometry and algebra. They show how representation theory interacts with multilinear algebra.

106.25 Regular Representation

Let $G$ be a finite group.

The regular representation is the representation of $G$ on the vector space with basis vectors

\{e_g:g\in G\}.

The action is

h e_g = e_{hg}.

Thus each group element permutes the basis according to left multiplication.

The regular representation is important because it contains all irreducible representations of $G$ over suitable fields.

It is one of the main tools for understanding finite group representations.

106.26 Permutation Representations

Suppose $G$ acts on a finite set $X$ .

Let $V$ be the vector space with basis

\{e_x:x\in X\}.

Define

g e_x=e_{gx}.

This gives a representation called a permutation representation.

Permutation representations translate actions on finite sets into linear algebra.

They appear in combinatorics, group actions, graph symmetry, and spectral methods.

106.27 Decomposing a Permutation Representation

Let $G=S_3$ act on

X=\{1,2,3\}.

The associated permutation representation acts on $\mathbb{R}^3$ by permuting coordinates.

The line

L=\operatorname{span} \left\{ \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \right\}

is invariant. It is a copy of the trivial representation.

The plane

W= \left\{ \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} : x_1+x_2+x_3=0 \right\}

is also invariant.

Thus

\mathbb{R}^3=L\oplus W.

The representation decomposes into a one-dimensional trivial part and a two-dimensional standard part.

106.28 Why Representation Theory Matters

Representation theory is useful because linear algebra is highly developed.

Once an algebraic object acts on a vector space, we can use:

Linear-algebra tool	Representation-theoretic use
Eigenvalues	Study action of elements
Trace	Define characters
Invariant subspaces	Find subrepresentations
Direct sums	Decompose representations
Tensor products	Build new representations
Dual spaces	Study covariant structures
Matrices	Compute explicitly

Representation theory therefore turns symmetry into linear structure.

106.29 Applications

Representation theory appears in many fields.

Area	Role
Group theory	Study groups through matrices
Lie theory	Study infinitesimal symmetries
Quantum mechanics	Symmetry actions on state spaces
Chemistry	Molecular symmetry
Fourier analysis	Decomposition into frequency modes
Number theory	Automorphic representations
Geometry	Symmetry of spaces and tensors
Machine learning	Equivariant models

The same formal language applies because all these areas involve structure-preserving linear actions.

106.30 Summary

Representation theory studies algebraic structures by their actions on vector spaces.

Concept	Meaning
Representation	Homomorphism into linear transformations
Representation space	Vector space being acted on
Subrepresentation	Invariant subspace
Irreducible representation	Representation with no nontrivial subrepresentations
Intertwiner	Linear map preserving the action
Character	Trace of the representing matrices
Tensor product representation	Combined action on $V\otimes W$
Faithful representation	Injective action

The basic principle is that abstract algebra becomes more tractable when expressed through linear transformations. Representation theory uses matrices and vector spaces to study symmetry, structure, decomposition, and invariance.