Group Representations | Matrices, Invariant Subspaces and Equivalence

The representation contract

Let G be a group and V a finite-dimensional vector space over a field F. A representation is a homomorphism ρ:G→GL(V). In coordinates, after choosing a basis of V, this becomes a matrix representation ρ:G→GL(n,F).

The contract is simple to state and easy to violate:

ρ(gh)=ρ(g)ρ(h).

Every calculation must remain compatible with the multiplication inside G. This means you cannot assign convenient matrices independently to each group element. The group’s relations constrain the matrices.

If G is given by generators and relations, representation construction often reduces to a controlled algebra problem: choose matrices for the generators that satisfy those same relations. Once that is done, the representation extends to every group element.

Representations from generators and relations

Suppose G=⟨g₁,…,g_k | R₁,…,R_m⟩. To define a representation on Fⁿ, choose invertible matrices A₁,…,A_k and check that every relation R_j becomes the identity matrix when the generators are replaced by the A_i.

This is more efficient than defining ρ on every element separately. The group presentation acts like a compact specification. The matrices are a candidate implementation. The relations are the test suite.

Example: the cyclic group C₆

C₆ has presentation ⟨r | r⁶=e⟩. A representation is determined by an invertible matrix A satisfying A⁶=I. Then ρ(r^k)=A^k.

Several different choices are possible. Over C, a one-dimensional representation can send r to any sixth root of unity. Over R, a two-dimensional representation can send r to the matrix for rotation through 60°. These are not cosmetic variants. They are genuinely different representations living over different fields and dimensions.

Example: the dihedral group

The dihedral group D_n has a presentation

D_n=⟨r,s | rⁿ=e, s²=e, srs=r⁻¹⟩.

To build a representation, choose matrices R and S satisfying Rⁿ=I, S²=I and SRS=R⁻¹. The standard geometric action on the plane uses a rotation matrix for R and a reflection matrix for S. The third relation encodes the geometric fact that reflecting reverses the direction of rotation.

This example shows why relations are more than symbolic bookkeeping. They preserve the operational meaning of the symmetry.

Matrix representations are coordinate descriptions

The representation itself is the action on V. Matrices appear only after we choose a basis. This distinction prevents a common misunderstanding: matrix entries are not intrinsic data.

Let B and B′ be two bases of V, with change-of-basis matrix P. If A_g is the matrix of ρ(g) in basis B, then the matrix in basis B′ is

A′_g=P⁻¹A_gP.

Every group element is conjugated by the same P. Therefore traces, determinants, characteristic polynomials and eigenvalue multisets are preserved for each fixed g. More importantly, the entire family of matrices still carries the same group action.

Representation theory therefore asks: which properties survive simultaneous change of basis? Those are the properties worth treating as structural.

Invariant subspaces as the first decomposition test

A subspace W≤V is G-invariant if ρ(g)W⊆W for every g in G. Since ρ(g) is invertible, this is equivalent to ρ(g)W=W.

Finding an invariant subspace means finding a smaller vector space on which the whole group still acts. This is the first signal that the representation may split into simpler pieces.

A single matrix having an eigenvector is not enough. The subspace must be stable under every representing matrix. Representation theory is about simultaneous structure.

Worked example: the permutation action of S₃ on F³

Let S₃ permute the coordinates of vectors in F³. The vector u=(1,1,1) is fixed by every permutation. Therefore U=span{u} is invariant.

Now consider

W={(x,y,z)∈F³ : x+y+z=0}.

Permuting coordinates preserves the sum x+y+z, so W is invariant. Provided the field behaves normally with respect to 3, we have F³=U⊕W. The original three-dimensional representation splits into a one-dimensional trivial representation and a two-dimensional standard representation.

In a basis beginning with u and continuing with two vectors spanning W, every permutation matrix becomes block diagonal:

[ 1   0 ]
[ 0  B_g]

The top-left 1 records the trivial action on U. The 2×2 block B_g carries the standard representation on W. The change of basis does not create the decomposition; it reveals a decomposition that was already present.

Reducibility, complete reducibility and splitting

A representation is reducible if it has a nonzero proper invariant subspace. But reducible does not automatically mean it is a direct sum of invariant subspaces.

To split V as W⊕W′, we need an invariant complement W′. For finite groups over a field whose characteristic does not divide |G|, Maschke’s theorem guarantees such complements. Over fields of modular characteristic, they may not exist.

This difference is central. Semisimple representation theory is largely about decomposing into irreducibles. Modular representation theory must also confront extensions, indecomposable representations and invariant subspaces that do not admit invariant complements.

Averaging: the idea behind Maschke’s theorem

Suppose W is an invariant subspace of V and we have any linear projection P:V→W. The projection need not respect the group action. For a finite group, we can average its conjugates:

P_G=(1/|G|) Σ_{g∈G} ρ(g)Pρ(g)⁻¹.

The averaged operator commutes with the group action. Its kernel becomes an invariant complement to W. The division by |G| is why the characteristic condition matters.

This averaging technique is a recurring representation-theoretic move: start with an arbitrary object and symmetrise it until it respects the group.

Equivalent representations

Representations (V,ρ) and (W,σ) are equivalent if there exists an invertible linear map T:V→W satisfying Tρ(g)=σ(g)T for every g.

This says T translates one action into the other without breaking symmetry. If V=W=Fⁿ, then σ(g)=Tρ(g)T⁻¹. In other words, equivalent matrix representations are simultaneously similar.

A single pair of matrices being similar is not enough. One fixed T must work for every group element.

A practical equivalence checklist

• Do the representations have the same dimension?
• Do corresponding group elements have the same traces?
• Do they have the same characters?
• Do kernels match?
• Do fixed-space dimensions match?
• Can you construct an invertible intertwiner?

For finite-dimensional complex representations of finite groups, equality of characters is enough to guarantee equivalence. Before character theory is available, however, direct intertwiner calculations remain instructive.

Intertwiners and the commutant

An intertwiner T:V→W satisfies Tρ(g)=σ(g)T for all g. If V=W and ρ=σ, such maps form the endomorphism algebra End_G(V), sometimes called the commutant or centraliser algebra of the representation.

The commutant measures how many linear transformations are compatible with the group action. For an irreducible complex representation, Schur’s lemma says every G-endomorphism is a scalar multiple of the identity.

This result is deceptively powerful. If an operator commutes with an irreducible symmetry action, there is almost no freedom left. In physics, this helps explain degeneracies and selection rules. In algebra, it constrains homomorphisms between irreducible modules and underpins character orthogonality.

Schur’s lemma in working form

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

• Any nonzero intertwiner T:V→W is an isomorphism.
• If V and W are not equivalent, every intertwiner is zero.
• Every intertwiner T:V→V is a scalar multiple of the identity.

The proof begins from a simple observation: kerT and imT are invariant subspaces. Irreducibility forces each to be either zero or the whole relevant space. The scalar conclusion follows by subtracting an eigenvalue multiple of the identity.

This illustrates a powerful habit: use invariant subspaces as a diagnostic tool. Whenever a linear map respects the group action, its kernel and image carry representation-theoretic information automatically.

Block diagonalisation and simultaneous structure

For one matrix, diagonalisation asks for a basis of eigenvectors. For a representation, the analogue is more subtle because we are dealing with an entire family of matrices.

If V decomposes as V₁⊕⋯⊕V_k into invariant subspaces, then there is a basis adapted to that decomposition in which every ρ(g) is block diagonal with the same block pattern. The individual blocks describe the action on the invariant pieces.

Representation theory is therefore not mainly about diagonalising each matrix separately. It is about finding a basis that reveals the common structure shared by all matrices in the image of G.

The trivial representation and fixed vectors

The trivial representation sends every g∈G to the identity. It may look uninteresting, but its multiplicity inside a representation measures the size of the fixed-vector space

V^G={v∈V : ρ(g)v=v for all g∈G}.

Fixed vectors represent quantities unchanged by the entire symmetry group. In geometry they may correspond to symmetry axes or invariant directions. In averaging problems they capture the fully symmetric component. In physics they can represent states left unchanged by a symmetry.

The sign representation

For the symmetric group S_n, the sign representation is one-dimensional: even permutations act by +1 and odd permutations act by −1.

The sign representation is important because it records orientation parity in the permutation. It appears naturally in determinants and alternating tensors. It is also a compact example of how a large non-abelian group can have a simple one-dimensional quotient representation.

The standard representation of S_n

S_n acts on Fⁿ by permuting coordinates. The line spanned by (1,…,1) is invariant and trivial. Its complementary hyperplane

x₁+⋯+x_n=0

is also invariant. This (n−1)-dimensional action is called the standard representation. For suitable fields it is irreducible.

The construction is worth remembering because it shows a common design pattern: begin with a natural permutation representation, identify an obvious trivial component, and quotient or complement it to obtain a more informative irreducible action.

Direct sums and multiplicity spaces

If V≅m₁V₁⊕⋯⊕m_rV_r, where the V_i are pairwise inequivalent irreducibles, then m_i is the multiplicity of V_i. The representation may contain several copies of the same irreducible action.

These multiplicities matter because they describe how many independent channels of the same symmetry type occur. Character theory will calculate them through inner products without requiring us to construct every invariant subspace explicitly.

Tensor products and new matrix actions

Given representations ρ on V and σ on W, define a representation on V⊗W by (ρ⊗σ)(g)=ρ(g)⊗σ(g).

If A_g and B_g are the representing matrices, the tensor-product matrix is the Kronecker product A_g⊗B_g. The dimensions multiply. The resulting representation often decomposes into irreducibles in a nontrivial way.

This is one of the places where representation theory becomes combinatorial. Knowing the irreducible building blocks is only the beginning; understanding how they combine is another layer of structure.

Dual representations

The dual vector space V* inherits a representation defined by

(ρ*(g)φ)(v)=φ(ρ(g⁻¹)v).

In matrices, the dual action is represented by (A_g⁻¹)^T. Duality matters because linear functionals transform contragrediently to vectors. This becomes important in tensor calculus, invariant pairings, physics and the representation theory of Lie groups and Lie algebras.

Matrix coefficients

Fix v∈V and φ∈V*. The function g↦φ(ρ(g)v) is a matrix coefficient of the representation. In a basis, ordinary matrix entries are examples of matrix coefficients.

These functions connect representation theory to harmonic analysis. For finite groups and compact groups, matrix coefficients behave like structured frequency components. Orthogonality relations generalise familiar Fourier orthogonality.

Worked example: C₄ over the complex numbers

Let C₄=⟨r⟩. Over C, every irreducible representation is one-dimensional because C₄ is abelian. Choose i=e^{πi/2}. The four irreducibles send r respectively to 1, i, −1 and −i.

A two-dimensional representation might be

ρ(r) = [ i   0 ]
       [ 0  -i ]

This is visibly a direct sum of two one-dimensional irreducibles. But if we change basis, the matrix may no longer be diagonal. The representation remains reducible because invariant one-dimensional subspaces still exist even if the coordinates hide them.

Worked example: a non-faithful representation

Again take C₄. Define a one-dimensional real representation by ρ(r)=−1. Since (−1)⁴=1, this is valid. But r² also maps to +1, so kerρ={e,r²}.

The action cannot distinguish r² from e. It sees only the quotient C₄/{e,r²}≅C₂. This is a clean example of how kernel records lost information.

How to test a candidate invariant subspace

If G is generated by g₁,…,g_k, you do not need to test every group element. It is enough to verify that W is stable under the matrices ρ(g₁),…,ρ(g_k) and, where needed, their inverses. Stability under generators propagates to the whole group.

• Choose a basis for W.
• Apply each generator matrix to each basis vector.
• Check whether the resulting vectors remain in W.
• If yes for all generators, W is invariant.

This turns invariant-subspace detection into a finite linear algebra problem when the group presentation is finite.

How to search for an intertwiner

Suppose ρ and σ are n-dimensional representations and you suspect they are equivalent. Let T be an unknown n×n matrix. For each generator g impose

Tρ(g)=σ(g)T.

These are linear equations in the entries of T. Solve the system. If the solution space contains an invertible matrix, the representations are equivalent. If all intertwiners are singular, they are not equivalent.

This is an excellent computational bridge: an abstract equivalence problem becomes a linear system.

Representation matrices and computational verification

For finite groups, software can help verify relations, compute traces, test invariant subspaces and solve intertwining equations. But computation should preserve the mathematical contract.

• Store generators rather than every element when possible.
• Verify defining relations before trusting downstream calculations.
• Check invertibility over the intended field.
• Distinguish exact arithmetic from floating-point approximations.
• When using numerical eigenvectors, verify invariance rather than assuming it from one matrix.
• Keep the group operation and basis conventions explicit.

Representation theory is especially vulnerable to silent coordinate mistakes because several matrices must remain mutually consistent. A computation that is locally correct for one matrix can still violate the global group action.

Common failure modes

• Checking only orders: matrices may have the right individual orders yet fail mixed relations.
• Confusing eigenvectors with invariant subspaces: invariance must hold for the full group action.
• Changing basis inconsistently: the same basis change must conjugate every representing matrix.
• Assuming reducible means diagonal: a reducible representation may require a new basis, and in modular settings may not split completely.
• Ignoring the field: decomposition can change after extending or restricting scalars.
• Testing equivalence element by element with different transforms: one fixed intertwiner must work globally.

Practice: representation construction

1. A cyclic representation

Let C₃=⟨r | r³=e⟩. Does the matrix A=[[0,−1],[1,−1]] define a representation by r↦A?

Answer: yes. Direct multiplication gives A³=I, so the defining relation is satisfied.

2. Fixed space

For the S₃ permutation representation on R³, find all vectors fixed by every group element.

Answer: every coordinate must be equal, so V^{S₃}=span{(1,1,1)}.

3. Kernel from a one-dimensional action

Let S₃ act on R by the sign representation. What is the kernel?

Answer: the alternating subgroup A₃, because exactly the even permutations act as +1.

4. Equivalent diagonal forms

Suppose ρ(r)=diag(i,−i) and σ(r)=diag(−i,i) for C₄. Are they equivalent?

Answer: yes. The basis-swap matrix conjugates one to the other and therefore intertwines the whole representation.

5. Direct-sum trace

If χ_V(g)=2 and χ_W(g)=−1, what is the character value of V⊕W at g?

Answer: 1, because traces add on block diagonal direct sums.

A deeper way to read the matrices

When you look at a representing matrix, do not ask only how to multiply it. Ask what role it plays in the whole action.

• Which group element does it represent?
• Which relation must it satisfy with the other generator matrices?
• Which subspaces does the full family preserve?
• Which quantities are unchanged by change of basis?
• What part of the group lies in the kernel?
• Can the action be decomposed?
• What does its trace tell us?

This shift—from isolated matrix calculations to simultaneous structure—is the transition from linear algebra into representation theory.

From matrices to characters

Matrix representations contain a great deal of information, but explicit matrices can become cumbersome. Character theory compresses each matrix to its trace while preserving enough information, in the finite-group complex setting, to recover decomposition into irreducibles.

The key reason traces work is that they are stable under similarity and additive under direct sums. They therefore ignore coordinate accidents while recording decomposition data. The next guide develops the resulting orthogonality machinery.

Continue the Representation Mathematics series

• Representation Theory in Mathematics | From Symmetry to Linear Actions
• Character Theory | Traces, Orthogonality and Irreducible Decomposition
• Representation Theory Applications | Geometry, Quantum Systems, Number Theory and Computation
• Return to the BTT Mathematics Learning Hub