Character Theory | Traces, Orthogonality and Irreducible Decomposition

The character of a representation

Let ρ:G→GL(V) be a finite-dimensional representation. Its character is the function χ_ρ:G→C defined by

χ_ρ(g)=tr(ρ(g)).

The value at the identity is especially important:

χ_ρ(e)=dim V.

So a character begins by recording the representation’s dimension. But its other values tell us how each symmetry acts after all basis-dependent detail has been compressed into a trace.

Why trace is the right compression

If two matrix representations are equivalent, then for one fixed invertible T we have σ(g)=Tρ(g)T⁻¹. Similar matrices have the same trace. Therefore equivalent representations have the same character.

Trace also behaves perfectly with direct sums. If V and W have characters χ and ψ, then the direct-sum representation has character

χ_{V⊕W}=χ+ψ.

This is exactly what decomposition theory needs. If a representation is a direct sum of irreducibles, its character is the corresponding sum of irreducible characters with multiplicities.

Tensor products behave multiplicatively:

χ_{V⊗W}(g)=χ_V(g)χ_W(g).

So direct sums become addition and tensor products become multiplication at the character level. This turns representation-building operations into ordinary arithmetic on functions.

Characters are class functions

Two elements g and hgh⁻¹ are conjugate. Their representation matrices satisfy

ρ(hgh⁻¹)=ρ(h)ρ(g)ρ(h)⁻¹.

Therefore conjugate elements produce similar matrices and have the same trace. Hence

χ(hgh⁻¹)=χ(g).

A character is constant on conjugacy classes. That immediately reduces the amount of data needed. Instead of recording one value per group element, we need only one value per conjugacy class.

For a finite group over C, the number of inequivalent irreducible representations equals the number of conjugacy classes. This is why character tables are arranged with irreducible characters as rows and conjugacy classes as columns.

The inner product on class functions

For complex-valued class functions φ and ψ on a finite group G, define

⟨φ,ψ⟩=(1/|G|) Σ_{g∈G} φ(g) overline{ψ(g)}.

Because class functions are constant on conjugacy classes, this can be computed class by class:

⟨φ,ψ⟩=(1/|G|) Σ_C |C| φ(C) overline{ψ(C)}.

This inner product is the engine of finite-group character theory. Irreducible characters are orthonormal:

⟨χ_i,χ_j⟩=δ_{ij}.

That one statement lets us test irreducibility, calculate multiplicities and verify character tables.

Irreducibility test

If χ is the character of a finite-dimensional complex representation, then

⟨χ,χ⟩=1

if and only if the representation is irreducible.

More generally, if V decomposes as

V≅m₁V₁⊕⋯⊕m_rV_r,

then χ_V=m₁χ₁+⋯+m_rχ_r and

⟨χ_V,χ_V⟩=m₁²+⋯+m_r².

So the self-inner-product measures how many irreducible components are present, counted through squared multiplicities.

Multiplicity formula

Suppose χ is the character of V and χ_i is an irreducible character. The multiplicity of V_i inside V is

m_i=⟨χ,χ_i⟩.

This is one of the most efficient results in the subject. We do not need to find invariant subspaces directly. We can compute traces, form an inner product and read off how many copies of each irreducible occur.

Matrix family → trace function → inner products → irreducible multiplicities.

The character table of S₃

S₃ has three conjugacy classes:

• the identity class {e}, size 1;
• the three transpositions, size 3;
• the two 3-cycles, size 2.

It therefore has three irreducible complex characters: trivial, sign and standard.

Class              e    transposition    3-cycle
Size               1         3              2
-------------------------------------------------
trivial            1         1              1
sign               1        -1              1
standard           2         0             -1

Several structural facts are visible immediately. The dimensions are 1,1,2. Their squares sum to 1+1+4=6=|S₃|. The trivial and sign rows differ only on odd permutations. The standard character has value 0 on reflections/transpositions and −1 on 3-cycles.

Checking orthogonality in S₃

For the standard character χ=(2,0,−1), compute its norm:

⟨χ,χ⟩=(1/6)(1·|2|²+3·|0|²+2·|−1|²)=(1/6)(4+2)=1.

So the standard representation is irreducible.

Its inner product with the trivial character is

(1/6)(1·2·1+3·0·1+2·(−1)·1)=0.

Thus the standard representation contains no trivial subrepresentation. This agrees with the fact that the standard two-dimensional space x+y+z=0 has no nonzero vector fixed by all of S₃.

Worked decomposition: the permutation representation of S₃

The three-dimensional permutation representation has character equal to the number of fixed points of each permutation:

• identity fixes 3 points, so χ(e)=3;
• a transposition fixes 1 point, so χ(transposition)=1;
• a 3-cycle fixes 0 points, so χ(3-cycle)=0.

Therefore χ_perm=(3,1,0).

Take inner products with the irreducible rows:

• with trivial: (1/6)(3+3+0)=1;
• with sign: (1/6)(3−3+0)=0;
• with standard: (1/6)(6+0+0)=1.

Hence

permutation representation ≅ trivial ⊕ standard.

Character theory has recovered the invariant-line decomposition without constructing a basis.

Permutation characters and fixed points

If G acts on a finite set X and we form the permutation representation on the vector space with basis X, then the character value χ(g) equals the number of points of X fixed by g.

This is a beautiful bridge between combinatorics and linear algebra. A trace of a permutation matrix counts its diagonal 1s, which are exactly the fixed points of the permutation.

Consequently, questions about orbits, fixed points and symmetry can often be translated directly into character calculations.

The regular character

For the regular representation of a finite group G, the character is especially simple:

• χ_reg(e)=|G|;
• χ_reg(g)=0 for every g≠e.

The reason is that left multiplication by a nonidentity element permutes the basis elements without fixing any of them.

The multiplicity of an irreducible character χ_i inside the regular representation is

⟨χ_reg,χ_i⟩=χ_i(e)=dim V_i.

Thus each irreducible appears with multiplicity equal to its dimension. Taking total dimension gives the familiar identity

|G|=Σ_i (dim V_i)².

Column orthogonality

Row orthogonality says irreducible characters are orthonormal as class functions. There is also a column version relating values at conjugacy classes. Roughly, columns become orthogonal when weighted appropriately, and the squared norm of a column is tied to the size of the centraliser of a representative.

These relations help complete partially known character tables. If most irreducible characters are known, orthogonality can force the remaining entries.

Characters of duals, sums and tensor products

• Direct sum: χ_{V⊕W}=χ_V+χ_W.
• Tensor product: χ_{V⊗W}=χ_Vχ_W pointwise.
• Dual: χ_{V*}(g)=overline{χ_V(g)} for finite groups in unitary form.
• Multiple copies: χ_{mV}=mχ_V.

This arithmetic makes characters an efficient construction language. Build a new representation, write down its character through these rules, then decompose by taking inner products with irreducible characters.

Worked example: tensoring the standard representation of S₃ with itself

The standard character is χ=(2,0,−1). The tensor-square character is the pointwise square

χ²=(4,0,1).

Decompose using inner products.

• With trivial (1,1,1): (1/6)(4+0+2)=1.
• With sign (1,−1,1): (1/6)(4+0+2)=1.
• With standard (2,0,−1): (1/6)(8+0−2)=1.

Therefore

standard ⊗ standard ≅ trivial ⊕ sign ⊕ standard.

The dimensions check: 2×2=4 and 1+1+2=4.

Dimension checks are not optional

Character calculations are exact but can still be entered incorrectly. A simple dimension check catches many mistakes.

If χ decomposes as Σm_iχ_i, then evaluating at the identity gives

dim V=Σ_i m_i dim V_i.

Use this after every decomposition. If the dimensions do not match, the multiplicities cannot be correct.

Character values and eigenvalues

For an element g of finite order m, the matrix ρ(g) satisfies ρ(g)^m=I. Over C it is diagonalizable with eigenvalues among the m-th roots of unity. The character value χ(g) is the sum of those eigenvalues.

This constrains possible character values. They are not arbitrary complex numbers. They are sums of roots of unity and hence algebraic integers. Character theory therefore carries hidden arithmetic information.

One-dimensional characters

A one-dimensional representation is a homomorphism G→C×. Its character is the homomorphism itself. Since C× is abelian, every one-dimensional character kills the commutator subgroup [G,G].

Therefore one-dimensional representations are controlled by the abelianisation G/[G,G]. This gives a useful structural shortcut: to find one-dimensional characters, study the group’s largest abelian quotient.

Character tables as compressed structure maps

A character table is not just a collection of traces. It captures several layers at once:

• the number of conjugacy classes;
• the dimensions of irreducible representations;
• the behaviour of irreducibles on different symmetry types;
• orthogonality relations;
• tensor-product decomposition data;
• information about kernels of representations;
• often clues about normal subgroups and quotient structure.

Two nonisomorphic groups can sometimes share the same character table, so the table is not a complete invariant of the group. But it is an exceptionally efficient summary of complex representation theory.

Kernel of a character

For a representation of dimension n with character χ, an element g lies in the kernel exactly when ρ(g)=I. For finite groups in a unitary realisation, this is equivalent to χ(g)=n.

So a character can reveal whether a representation is faithful: the only class where the character equals the full dimension should be the identity class.

Restriction

If H≤G and V is a representation of G, we can restrict the action to H. The character simply restricts as a function: χ_V|_H.

An irreducible representation of G can become reducible when restricted to a subgroup. Studying how irreducibles break when moving down to subgroups produces branching rules and connects representation theories across a subgroup lattice.

Induction

Induction moves in the opposite direction. Starting from a representation of a subgroup H, one constructs a representation of G. The induced representation is larger and encodes how the subgroup action propagates across cosets of H.

Characters make induction calculable. Frobenius reciprocity connects restriction and induction through inner products:

⟨Ind_H^G ψ, χ⟩_G = ⟨ψ, Res_H^G χ⟩_H.

This is a deep example of mathematical adjunction in working form: multiplicities computed after moving upward agree with multiplicities computed after moving downward.

Burnside-style counting connection

For a permutation representation, χ(g)=|Fix(g)|. The average character value is therefore

(1/|G|)Σ_g |Fix(g)|.

This equals the inner product of the permutation character with the trivial character, which counts the multiplicity of the trivial representation. That multiplicity is the dimension of the fixed-vector space and, for a permutation action, equals the number of orbits.

This recovers Burnside’s orbit-counting lemma from character-theoretic language. A combinatorial counting theorem appears as a statement about the trivial component inside a permutation representation.

Common mistakes in character calculations

• Ignoring class sizes: the inner-product sum must weight each conjugacy class by its size.
• Forgetting complex conjugation: the second character is conjugated in the inner product.
• Using dimensions instead of character values: χ(e) is the dimension, but other columns require actual traces.
• Assuming any class function is a character: genuine characters satisfy strong integrality and positivity constraints.
• Skipping the dimension check: multiplicities must reconstruct the original dimension.
• Confusing character equality with matrix equality: equal characters imply equivalent complex representations, not identical matrices in the same basis.

A character-table workflow

• List the conjugacy classes and their sizes.
• Find obvious one-dimensional characters.
• Use the number of conjugacy classes to know how many irreducible rows are required.
• Use Σ d_i²=|G| to constrain irreducible dimensions.
• Construct natural representations and compute their characters.
• Decompose them using inner products.
• Use row and column orthogonality to fill missing values.
• Check dimensions, norms and orthogonality before treating the table as complete.

Practice set

1. Character at the identity

A representation has character χ with χ(e)=5. What is dimV?

Answer: 5.

2. Direct sum

At an element g, χ_V(g)=2 and χ_W(g)=−3. Find χ_{V⊕W}(g).

Answer: −1.

3. Tensor product

At g, χ_V(g)=2 and χ_W(g)=−3. Find χ_{V⊗W}(g).

Answer: −6.

4. Irreducibility norm

A character has self-inner-product 1. What can you conclude?

Answer: the corresponding complex representation is irreducible.

5. Multiplicity

If ⟨χ,χ_i⟩=3, what does the 3 mean?

Answer: the irreducible representation with character χ_i occurs three times in the decomposition.

6. Regular representation

A group has an irreducible representation of dimension 4. How many copies of it appear in the regular representation?

Answer: four copies.

7. Permutation character

If a group element fixes exactly seven points in a permutation action, what is the corresponding permutation-character value?

Answer: 7.

Why character theory is more than a shortcut

It is tempting to think of characters as a computational trick for avoiding large matrices. They are more important than that.

Character theory identifies the right level of information for a whole class of structural questions. Matrix entries depend on basis. Eigenvectors may shift under coordinates. But traces on conjugacy classes survive equivalence and interact cleanly with decomposition. The character is therefore a deliberately compressed representation of a representation.

Good mathematics does not merely calculate more. It finds a representation in which the relevant structure becomes cheaper to see.

Beyond finite groups

Character ideas extend far beyond finite groups. Compact groups have characters tied to harmonic analysis. Lie groups and Lie algebras have rich highest-weight character theories. Number theory uses characters in several different senses, including Dirichlet characters and characters attached to representations. Modern representation theory often replaces finite character tables with more sophisticated functions, distributions or categorical invariants.

The finite-group theory remains an ideal training ground because it displays the full architecture—symmetry, linearisation, invariance, decomposition, orthogonality and reconstruction—in a setting where explicit examples can be completed by hand.

Continue the Representation Mathematics series

• Representation Theory in Mathematics | From Symmetry to Linear Actions
• Group Representations | Matrices, Invariant Subspaces and Equivalence
• Representation Theory Applications | Geometry, Quantum Systems, Number Theory and Computation
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