Representation Theory in Mathematics | From Symmetry to Linear Actions

Why representation theory exists

Many mathematical structures are easy to define but difficult to inspect directly. A finite group, for example, may be given by generators and relations, a multiplication table or a symmetry description. Those forms tell us what the group is, but not always which internal features matter most.

Linear algebra gives us an unusually rich diagnostic environment. Once an abstract operation is represented by a matrix, we can ask for determinants, traces, eigenvalues, eigenspaces, invariant subspaces and decompositions. We can compare two actions by changing basis. We can combine representations by direct sums and tensor products. We can sometimes break a large problem into smaller irreducible pieces.

Abstract symmetry → linear action → computable structure → decomposition → interpretation.

Representation theory therefore sits at a junction. It is algebra because groups, rings and algebras are central. It is linear algebra because vector spaces and matrices carry the action. It reaches geometry because symmetry acts on spaces. It reaches number theory because arithmetic objects have symmetry. It reaches physics because quantum states and physical symmetries are naturally linear. It reaches computation because finite-dimensional representations can be stored, transformed and analysed algorithmically.

The core definition

Let G be a group and let V be a vector space over a field F. A representation of G on V is a group homomorphism

ρ : G → GL(V).

Here GL(V) is the group of invertible linear transformations from V to itself. The homomorphism condition says

ρ(gh) = ρ(g)ρ(h) for all g,h in G.

It also forces ρ(e) to be the identity transformation and ρ(g⁻¹) to be the inverse of ρ(g). The group law survives the translation. This preservation is the whole point. If multiplication in G were not respected, the assigned matrices might be interesting objects, but they would not constitute a representation.

If V has a chosen basis and dimension n, every linear transformation can be written as an n×n matrix. The representation may then be viewed as a homomorphism from G into GL(n,F). Different bases produce different matrices, but they can describe the same underlying representation.

A first example: the symmetries of an equilateral triangle

An equilateral triangle has six rigid symmetries: three rotations and three reflections. These symmetries form the dihedral group D₃, which is isomorphic to the permutation group S₃.

Place the triangle with its centre at the origin in the plane. A rotation by 120° acts on every vector in the plane by a 2×2 rotation matrix. A reflection acts by a 2×2 reflection matrix. Composition of geometric symmetries corresponds to matrix multiplication. We have therefore obtained a two-dimensional real representation.

The important idea is not the particular coordinates. The triangle gives a physical picture; the matrices give a linear model; the group gives the abstract multiplication law. Representation theory lets all three descriptions refer to the same symmetry system.

If we rotate our coordinate axes, every representing matrix changes by conjugation with the basis-change matrix. The representation has not changed in substance. Only its coordinates have changed. This is why equivalence of representations matters so much: representation theory is interested in structure that survives a change of basis.

Group actions and representations are related but not identical

A group action on a set X assigns each g in G to a permutation of X. A representation is a group action on a vector space where every action map is linear.

• A group acting on the vertices of a polygon gives a permutation action.
• A group acting on coordinate vectors through matrices gives a linear representation.
• A permutation action can often be converted into a permutation representation on a vector space with basis vectors indexed by the set being permuted.

This conversion is one of the simplest bridges into representation theory. Suppose G permutes n objects. Let V have basis e₁,…,eₙ. If g sends object i to object j, define ρ(g)eᵢ=eⱼ. Each ρ(g) is represented by a permutation matrix. The original combinatorial symmetry becomes a linear action.

Worked example: a permutation representation

Consider the transposition σ=(12) acting on three symbols {1,2,3}. On the vector space with basis e₁,e₂,e₃, define

• ρ(σ)e₁=e₂,
• ρ(σ)e₂=e₁,
• ρ(σ)e₃=e₃.

Relative to the ordered basis (e₁,e₂,e₃), the matrix is

[0 1 0]
[1 0 0]
[0 0 1]

Squaring this matrix gives the identity, exactly as σ²=e in the group. The matrix has eigenvalue 1 on vectors unchanged by the swap and eigenvalue −1 on the direction e₁−e₂. Already we can see representation theory turning a permutation into invariant geometric information.

Faithful and non-faithful representations

A representation is faithful when different group elements always produce different linear transformations. Equivalently, the kernel of ρ contains only the identity.

Faithfulness matters when we want the linear model to retain all distinctions present in the original group. But non-faithful representations are not defective. They may deliberately ignore part of the structure and reveal a useful quotient.

If several group elements act identically on V, the representation cannot distinguish them. Those elements lie in the kernel. The image of the representation is then isomorphic to G/kerρ. This tells us exactly what portion of the group is visible through that representation.

A representation is not merely a picture of a group. It is a measurement of which parts of the group become visible through a chosen linear action.

Invariant subspaces: where the action closes on itself

Let W be a subspace of V. We call W invariant under the representation if ρ(g)w belongs to W for every g in G and every w in W.

An invariant subspace is a smaller region of V that the entire group action respects. If such a nontrivial proper subspace exists, the representation may contain smaller pieces that can be studied separately.

Return to the permutation representation of S₃ on F³. The line spanned by e₁+e₂+e₃ is invariant because every permutation leaves this sum unchanged. There is also a two-dimensional subspace consisting of vectors whose coordinates sum to zero. That subspace is invariant too. Thus the three-dimensional permutation representation contains a one-dimensional trivial piece and a two-dimensional standard piece.

This is the beginning of decomposition theory. Large representations become more understandable when we identify invariant pieces that do not interact arbitrarily.

Reducible and irreducible representations

A nonzero representation is irreducible if its only invariant subspaces are {0} and V itself. If it has another invariant subspace, it is reducible.

Irreducible representations play a role analogous to primes in arithmetic or atoms in a decomposition problem. They are not always literally indivisible in every category or every field, but they are the basic pieces under the relevant notion of invariant linear structure.

For finite groups over the complex numbers, Maschke’s theorem tells us something powerful: every finite-dimensional representation decomposes as a direct sum of irreducible representations. This semisimplicity depends on the characteristic of the field not dividing the group order. The theorem is therefore a reminder that representations depend not only on the group but also on the field over which the vector space is defined.

Why the field matters

The same group can have representation theories that behave differently over R, C and finite fields. A matrix may have eigenvalues over C that do not exist over R. An irreducible real representation may split after extending scalars to C. In modular representation theory, where the characteristic divides the group order, complete reducibility can fail.

This is a useful general mathematical lesson: changing the ambient number system can change what counts as a decomposition, what eigenvalues exist and which structural statements remain true.

Equivalent representations and change of basis

Two representations ρ:G→GL(V) and σ:G→GL(W) are equivalent if there is an invertible linear map T:V→W such that

Tρ(g)=σ(g)T for every g in G.

If V and W are the same coordinate space, this becomes σ(g)=Tρ(g)T⁻¹. The matrices can look different while encoding the same action in different bases.

This is one reason raw matrix entries are rarely the final invariant. Representation theory searches for features that survive equivalence: dimension, character, multiplicities of irreducible constituents, determinant patterns, fixed-space dimensions and other structural quantities.

Intertwiners: maps that respect the action

A linear map T:V→W between two representations is called an intertwining map, or G-map, if Tρ(g)=σ(g)T for all g. Such a map respects the symmetry action.

Intertwiners are the correct morphisms in the category of representations. Their kernels and images are invariant subspaces. An invertible intertwiner proves equivalence. Schur’s lemma describes intertwiners between irreducible representations and becomes one of the foundational tools behind character theory and the structure of irreducible modules.

Direct sums: combining independent pieces

If V and W carry representations ρ and σ, the direct sum V⊕W carries a representation defined by

(ρ⊕σ)(g)(v,w)=(ρ(g)v,σ(g)w).

In a compatible basis, the representing matrices are block diagonal. This makes decomposition visible. A reducible representation that splits as a direct sum can be reorganised so that each invariant component occupies its own block.

Representation theory often asks not only whether a decomposition exists, but how many copies of each irreducible representation occur. Those multiplicities are among the quantities character theory can compute efficiently.

Tensor products: combining actions multiplicatively

The tensor product V⊗W carries the representation

(ρ⊗σ)(g)=ρ(g)⊗σ(g).

Tensor products create new representations from old ones. They are central in physics, multilinear algebra and the internal structure of representation categories. Unlike direct sums, tensor products mix dimensions multiplicatively: if V has dimension m and W has dimension n, then V⊗W has dimension mn.

A major practical question is how a tensor product decomposes into irreducibles. In finite-group theory this can be attacked with characters. In Lie theory it leads to rich combinatorics and highest-weight methods.

The regular representation

Every finite group G acts on the vector space with basis vectors indexed by its own elements. Left multiplication sends e_h to e_{gh}. This gives the left regular representation.

The regular representation is large—its dimension is |G|—but it contains remarkable information. Over C, every irreducible representation appears inside it, and the multiplicity of an irreducible equals its dimension. Consequently, if the irreducible dimensions are d₁,…,d_r, then

|G| = d₁² + d₂² + ··· + d_r².

This equation is a compact structural fingerprint. It constrains the possible irreducible dimensions before we know the actual matrices.

Worked example: irreducible dimensions of S₃

The group S₃ has order 6 and three conjugacy classes, so over C it has three inequivalent irreducible representations. Two one-dimensional representations are easy to identify: the trivial representation and the sign representation.

If the third irreducible has dimension d, the regular-representation dimension formula gives

6 = 1² + 1² + d².

Therefore d²=4 and d=2. Without constructing every matrix explicitly, we already know the irreducible dimension pattern: 1,1,2.

The standard two-dimensional representation can be obtained from the three-dimensional permutation representation by removing the invariant line spanned by e₁+e₂+e₃. This is a typical representation-theoretic move: construct a natural large action, locate an obvious invariant piece, then study the complementary structure.

Characters: the next compression layer

For a finite-dimensional representation ρ, its character is the function χ:G→F given by

χ(g)=tr(ρ(g)).

Characters compress matrix information into traces. That sounds like a severe loss, yet for finite groups over C the character determines the representation up to equivalence. Characters are constant on conjugacy classes, and their inner products detect irreducibility and multiplicity.

The next guide develops this systematically: Character Theory | Traces, Orthogonality and Irreducible Decomposition.

Three levels of seeing the same symmetry

• Geometric level: rotate or reflect an object.
• Abstract group level: retain only the composition law between operations.
• Representation level: realise the group law by linear transformations.

Each level forgets some information and exposes other information. Geometry may show intuition. The abstract group strips away irrelevant coordinates. The representation restores calculability without returning to the original physical object.

This movement—abstract, represent, calculate, interpret—is one of the most reusable patterns in advanced Mathematics.

What a representation preserves

A good way to study any representation is to ask what survives the translation.

• The identity becomes the identity transformation.
• Products become products of transformations.
• Inverses become inverse transformations.
• Conjugate group elements become similar matrices and therefore have the same trace.
• Invariant subspaces expose internally stable components of the action.
• Kernel and image reveal what is invisible and visible in the chosen model.

This preservation logic is representation theory’s control system. Matrices are useful because they obey the group law, not merely because matrices are familiar computational objects.

Common misconceptions

“A representation is just a drawing.”

In this subject, representation has a precise algebraic meaning: a homomorphism into a group of invertible linear transformations.

“Different matrices mean different representations.”

Not necessarily. A change of basis conjugates all representing matrices simultaneously and yields an equivalent representation.

“Faithful is always better.”

A faithful representation retains every group distinction, but a non-faithful representation can intentionally factor through a quotient and isolate the structure relevant to a specific problem.

“Irreducible means one-dimensional.”

No. Irreducible means there is no nonzero proper invariant subspace. Irreducible representations can have any dimension permitted by the group and field.

A learning sequence for representation theory

• Become fluent with vector spaces, bases, linear maps, eigenvalues and change of basis.
• Learn groups, subgroups, normal subgroups, quotients, homomorphisms and conjugacy.
• Build concrete representations from symmetries and permutations.
• Identify kernels, invariant subspaces and decompositions.
• Study irreducibility, Schur’s lemma and Maschke’s theorem.
• Move to characters and orthogonality.
• Use tensor products and induced constructions to create new representations.
• Extend to Lie groups, Lie algebras, modular theory or arithmetic applications as appropriate.

Mini practice set

1. Homomorphism check

Suppose G=C₄=⟨r | r⁴=e⟩. Let ρ(r) be the 2×2 rotation matrix through 90°. Explain why this determines a representation of C₄ on R².

Answer: the 90° rotation matrix A satisfies A⁴=I, so the defining relation r⁴=e is respected. Set ρ(r^k)=A^k. Then products in C₄ correspond to matrix products.

2. Kernel

Let C₄ act on a one-dimensional real vector space by sending r to −1. What is the kernel?

Answer: ρ(r)=−1, ρ(r²)=1, ρ(r³)=−1, ρ(e)=1. The kernel is {e,r²}. The representation factors through C₂.

3. Invariant line

In the permutation representation of S₃ on F³, why is the line spanned by (1,1,1) invariant?

Answer: permuting coordinates leaves (1,1,1) unchanged, so every group element maps the line to itself.

4. Equivalence

If σ(g)=Tρ(g)T⁻¹ for one fixed invertible T and every g, what has changed?

Answer: only the coordinate basis. The representations are equivalent.

5. Dimension constraint

A finite group of order 8 has four one-dimensional irreducible complex representations. If there is exactly one other irreducible representation, what is its dimension?

Answer: 8=4·1²+d², so d²=4 and d=2.

Representation theory as a mathematical habit

The subject teaches a general strategy that extends far beyond groups. When an abstract structure is difficult to inspect, seek a representation in a better-understood environment. Preserve the operations that matter. Identify which features are invariant under changes of coordinates. Decompose the representation into stable pieces. Then return to the original structure with information that was difficult to see before.

That strategy appears across modern Mathematics: group representations, modules over rings, linear representations of Lie algebras, Galois representations, harmonic analysis, operator theory and quantum mechanics all use versions of the same move.

Representation is not a retreat from abstraction. It is a controlled way of making abstraction operational.

Continue the Representation Mathematics series

• Group Representations | Matrices, Invariant Subspaces and Equivalence
• Character Theory | Traces, Orthogonality and Irreducible Decomposition
• Representation Theory Applications | Geometry, Quantum Systems, Number Theory and Computation
• Return to the BTT Mathematics Learning Hub