FOURIER ANALYSIS ON FINITE NON-ABELIAN GROUPS

TERENCE TAO

Abstract. Some notes on the Fourier transform on finite non-abelian groups G. These notes are aimed at readers who are familiar with Fourier analysis on abelian groups such as R, Z, and T, but are not familiar with representation theory of non-abelian groups.

1. Hilbert spaces

In these notes, when we say that V is a Hilbert space, we mean that V is a finite dimensional complex vector space equipped with an inner product ⟨v, w⟩ which is Hermitian (linear in v, anti-linear in w) and positive definite.

Given any linear transformation T : V → V from a Hilbert space to itself, we can define its trace tr(T) by the formula

tr(T) := ∑_{e}⟨e, T e⟩

where e ranges over some orthonormal basis of V; it is easy to see that this definition is independent of the choice of basis. Observe that trace is linear and that tr(ST) = tr(T S).

If V, W are finite dimensional complex Hilbert spaces, we define HS(V → W) to be the space of all Hilbert-Schmidt operators from V to W; since V, W are finitedimensional, HS(V → W) is just the same as Hom(V → W). We observe that HS(V → W) is also a Hilbert space, with the Hilbert-Schmidt inner product

⟨S, T ⟩_{HS(V →W)} := tr(ST^{∗}).

where S^{∗} is of course the adjoint of S. We abbreviate HS(V → V) as HS(V).

We use I_{V} : V → V to denote the identity operator on V; note that tr(I_{V}) = dim(V).

We use U(V) ⊂ HS(V) to denote the space of unitary operators on V, i.e. those operators T ∈ HS(V → V) such that T T^{∗} = T^{∗}T = I_{V}. This forms a group with identity I_{V}.

2. Finite groups

Let G be a finite group with group operation ∘, identity id, and cardinality |G|. We give this group normalized counting measure dx:

∫_{G} f(x) dx := \frac{1}{|G|} ∑_{x∈G} f(x),

and then define L^{2}(G) to be the Hilbert space of functions f : G → C endowed with the inner product

⟨f, g⟩ := ∫_{G} f(x)\overline{g(x)} dx.

Note that this inner product is always finite since G is finite.

We note a special type of function in L^{2}(G): the Dirac delta functions. Specifically, for every x ∈ G, we define δ_{x} ∈ L^{2}(G) by duality, setting

⟨f, δ_{x}⟩ = f(x) for all f ∈ L^{2}(G).  (1)

Equivalently, we can define δ_{x}(y) := |G| when x = y, and δ_{x}(y) := 0 when x ≠ y. Note that δ_{x} is indeed in L^{2}(G) because G is finite. Indeed, the functions {δ_{x} : x ∈ G} form a basis for L^{2}(G), thanks to the reproducing formula

f = ∫_{G}⟨f, δ_{x}⟩δ_{x} dx.

Given any y ∈ G, we can define the translation operator τ_{y} : L^{2}(G) → L^{2}(G) by the formula

τ_{y}δ_{x} := δ_{y∘x} for all x ∈ G

and then extending τ_{y} by linearity. In other words, we have

τ_{y}f(x) := f(y^{−1} ∘ x) for all f ∈ L^{2}(G).

Observe that τ_{y} acts unitarily on L^{2}(G) and we have the homomorphism property

τ_{x}τ_{y} = τ_{x∘y}

(this is best seen by first checking this on Dirac delta functions). In particular, τ_{id} is the identity, and τ_{x^{−1}} is the inverse of τ_{x}.

We define the operation of convolution ∗ : L^{2}(G) × L^{2}(G) → L^{2}(G) by defining δ_{x} ∗ δ_{y} := δ_{x∘y} and extending by bilinearity. Thus

f ∗ g(x) = ∫_{G} f(y)g(y^{−1} ∘ x) dy.

This operation turns^{1} L^{2}(G) into an algebra, called the convolution algebra; this algebra is commutative if and only if the group G is. One of the main purposes of the Fourier transform is to recast this algebra in a more tractable form.

3. Representations

By a (unitary) representation ρ of G, we mean a Hilbert space V_{ρ}, together with a homomorphism ρ : G → U(V). In other words, for each group element x we assign a unitary operator ρ(x) : V_{ρ} → V_{ρ} such that ρ(x)ρ(y) = ρ(xy).

Examples of representations include the zero representation 0, with V_{0} = {0} and 0(x) := 0; the trivial representation ρ_{0}, with V_{ρ_{0}} = C and ρ_{0}(x) = IC; and the regular representation τ, with V_{τ} := L^{2}(G) and τ(x) := τ_{x}.

A morphism ϕ : ρ_{1} → ρ_{2} between two representations is any unitary map ϕ : V_{ρ_{1}} → V_{ρ_{2}} such that we have the intertwining relationship

ρ_{2}(x)ϕ = ϕρ_{1}(x) for all g ∈ G.

If the morphism has an inverse, we say it is an isomorphism.

We say that two representations ρ_{1}, ρ_{2} are isomorphic, and write ρ_{1} ≡ ρ_{2}, if there is an isomorphism from ρ_{1} to ρ_{2}. This is clearly an equivalence relation.

1In the case of infinite groups, the relevant algebra would be L^{1}(G) rather than L^{2}(G). For finite groups, of course, these spaces are the same.

A representation ρ gives a map from G to U(V_{ρ}), which in turn is a subset of HS(V_{ρ}). This induces a map from L^{2}(G) to HS(V_{ρ}) by defining ρ(δ_{x}) := ρ(x) and extending by linearity. Thus

ρ(f) := ∫_{G} f(x)ρ(x) dx.  (2)

For any representation ρ, we observe the identity

ρ(f ∗ g) = ρ(f)ρ(g);  (3)

this is easiest to see by checking first on delta functions. (This gives the first hint as to what the Fourier transform on G should be; but more on this later.)

4. The regular representation

Perhaps the most important representation for Fourier analysis is the regular representation τ, defined in the previous section. Observe that for any f ∈ L^{2}(G), the Hilbert-Schmidt operator τ(f) is the same as convolution by f:

τ(f)g = f ∗ g.  (4)

This is easiest seen by first verifying the case when f, g are both delta functions, and then using bilinearity.

The map f ↦ τ(f) is a linear map from L^{2}(G) to HS(L^{2}(G)). This map is in fact an isometry:

Lemma 4.1. Let f ∈ L^{2}(G). Then we have

∥τ(f)∥_{HS(L^{2}(G))} = ∥f∥_{L^{2}(G)}.

In particular, this implies the regular representation is faithful (i.e. the map f ↦ τ(f) is injective). Proof Squaring both sides, we reduce to showing that

tr(τ(f)^{∗}τ(f)) = ∫_{G} |f(x)|^{2} dx.

We have the easily verified identity tr(T) = ∫_{x}⟨δ_{x}, T δ_{x}⟩ dx for any T ∈ HS(L^{2}(G)). Thus

∥τ(f)∥_{HS(L^{2}(G))}^{2} = tr(τ(f)^{∗}τ(f))
= ∫_{G}⟨τ(f)^{∗}τ(f)δ_{x}, δ_{x}⟩ dx
= ∫_{G}⟨τ(f)δ_{x}, τ(f)δ_{x}⟩ dx
= ∫_{G} ∥f ∗ δ_{x}∥_{L^{2}(G)}^{2} dx
= ∫_{G} ∥f∥_{L^{2}(G)}^{2} dx
= ∥f∥_{L^{2}(G)}^{2}

as desired.

From (3) we have τ(f ∗ g) = τ(f)τ(g) (this is just a fancy way of saying convolution is associative). Thus the map f ↦ τ(f) is not only a Hilbert space isometry, it is also a homomorphism of algebras, mapping the convolution algebra of L^{2}(G) to some operator sub-algebra of HS(L^{2}(G)).

In some sense, the map f ↦ τ(f) already captures the essence of the Fourier transform. However, things are not yet satisfactory because this map f ↦ τ(f) is not onto; not every Hilbert-Schmidt operator on L^{2}(G) is a convolution operator. Thus we must discover exactly what subspace of HS(L^{2}(G)) is the image of τ.

In the abelian case, we know that there is a one-to-one correspondence between convolution operators and Fourier multipliers; to put it another way, the convolution operators are precisely those operators which are diagonal in the Fourier basis. In the non-abelian case, we will have a similar characterization, except that we will replace “diagonal” by “block-diagonal”. To do this we need a notion of direct sum and irreducible representation, to which we now turn.

5. Irreducibility

If V and W are Hilbert spaces, then V ⊕ W := {(v, w) : v ∈ V, w ∈ W } is also a Hilbert space, and if T : V → V and S : W → W are Hilbert-Schmidt, we can define the direct sum T ⊕ S : V ⊕ W → V ⊕ W as (T ⊕ S)(v, w) := (T v, Sw).

Given two representations ρ_{1} and ρ_{2}, we can define their direct sum ρ_{1} ⊕ ρ_{2} by defining V_{ρ_{1}⊕ρ_{2}} := V_{ρ_{1}} ⊕ V_{ρ_{2}} and (ρ_{1} ⊕ ρ_{2})(x) := ρ_{1}(x) ⊕ ρ_{2}(x). By linearity, we thus have that

(ρ_{1} ⊕ ρ_{2})(f) = ρ_{1}(f) ⊕ ρ_{2}(f)  (5)

Also observe that ρ_{1} ⊕ ρ_{2} ≡ ρ_{2} ⊕ ρ_{1}.

We call a representation ρ reducible if it can be written as ρ ≡ ρ_{1} ⊕ ρ_{2} for some non-zero representations ρ_{1}, ρ_{2}, and irreducible if it is non-zero and not reducible; we consider the zero representation to be neither reducible nor irreducible. We will use Γ to index the irreducible representations of G, up to isomorphism; thus every ξ ∈ Γ gives rise to an irreducible representation ρ_{ξ}, and ρ_{ξ} ≡ ρ_{ξ′} if and only if ξ = ξ′. We shall index Γ so that ρ_{0} is the trivial representation. We shall also abbreviate V_{ρ_{ξ}} as V_{ξ}.

Because we are requiring all our representations to be finite dimensional, every representation ρ can be decomposed as the sum of a finite number of irreducible representations,

ρ ≡ ⨁_{ξ∈Γ} ρ_{ξ}^{⊕c_{ξ}}

where the c_{ξ} are some non-negative integers (called the multiplicity of ρ_{ξ} in ρ), and ρ_{ξ}^{⊕c_{ξ}} is the direct sum of c_{ξ} copies of ρ_{ξ}. There is an issue as to whether the coefficients c_{ξ} are unique; they are, but we will prove this later when we derive an explicit formula for them.

In particular, the regular representation can be decomposed as

τ ≡ ⨁_{ξ∈Γ} ρ_{ξ}^{⊕m_{ξ}}  (6)

for some non-negative integers m_{ξ}. Note that since τ is finite dimensional, at most finitely many of the m_{ξ} are non-zero.

We now fix the m_{ξ} (if there is failure of uniqueness, we pick m_{ξ} arbitrarily; later on we will compute m_{ξ} explicitly and show that one does have uniqueness, but it is not needed for this present discussion). Now let f ∈ L^{2}(G). By (6) and (5) we see that

∥τ(f)∥_{HS(L^{2}(G))}^{2} = ∑_{ξ∈Γ} m_{ξ}∥ρ_{ξ}(f)∥_{HS(V_{ξ})}^{2}.

We rewrite this by defining a measure dξ on Γ by

∫_{Γ} f(ξ) dξ := ∑_{ξ∈Γ} m_{ξ}f(ξ).

We also rewrite ρ_{ξ}(f) as f̂(ξ), thus

f̂(ξ) = ∫_{G} f(x)ρ_{ξ}(x) dx  (7)

is a Hilbert-Schmidt operator on V_{ξ} for each ξ. Then, using Lemma 4.1, we have the Plancherel formula

∥f∥_{L^{2}(G)}^{2} = ∫_{Γ} ∥f̂(ξ)∥_{HS(V_{ξ})}^{2} dξ  (8)

for all f ∈ L^{2}(ξ).

We have thus defined a Fourier transform f ↦ f̂, which is an isometry from L^{2}(G) to ∫_{Γ} HS(V_{ξ}) dξ, where we use ∫_{Γ} HS(V_{ξ}) dξ to denote the Hilbert space obtained by taking the direct sum of all the HS(V_{ξ}) and endowing the resulting space with the inner product indicated (8). Also, from (3) we observe the familiar identity

{ \atop f ∗ g}(ξ) = f(ξ)g(ξ).  (9)

Thus the Fourier transform is an algebra homomorphism as well as an isometry. Next, we consider how to invert the Fourier transform. By depolarizing (8), we obtain the Parseval identity

⟨f, g⟩ = ∫_{Γ}⟨f̂(ξ), ĝ(ξ)⟩_{HS(V_{ξ})} dξ

for all f, g ∈ L^{2}(G). In particular, from (1) we have

f(x) = ∫_{Γ}⟨f̂(ξ), δ̂_{x}(ξ)⟩_{HS(V_{ξ})} dξ

for all x ∈ G. Since

δ̂_{x}(ξ) = ρ_{ξ}(x)  (10)

by definition, we thus have the Fourier inversion formula

f(x) = ∫_{Γ}⟨f̂(ξ), ρ_{ξ}(x)⟩_{HS(V_{ξ})} dξ.  (11)

From (9) and (11) we see that translations map to modulations on the Fourier transform side:

τ̂_{x}f(ξ) = { \atop δ_{x} ∗ f}(ξ) = ρ_{ξ}(x)f̂(ξ).  (12)

We have already derived many of the familiar properties of the Fourier transform. There are two major tasks left to do to obtain a truly satisfactory theory: first, we have to show that the Fourier transform is in fact surjective (we already know it is injective by (11)), and secondly we have to work out what the coefficients m_{ξ}. (There is also the question of how one locates all the irreducible representations Γ, but this leads of course into the field of representation theory, which we will not detail here.)

In order to address these two questions we need a fundamental tool in representation theory, namely Schur’s lemma.

6. Schur’s lemma

A basic fact about irreducible representations is that they have no non-trivial invariant spaces:

Lemma 6.1. Let ξ ∈ Γ, and let W be a subspace of V_{ξ} such that ρ_{ξ}(x)W ⊆ W for all x ∈ G (i.e. W is an invariant space of ρ_{ξ}). Then W = V_{ξ} or W = {0}.

Proof Suppose W is a proper subspace of V_{ξ}. Then W^{⊥}, the orthogonal complement of W in V_{ξ}, is also a proper subspace. Since ρ_{ξ}(x) is unitary and leaves W invariant, it also leaves W^{⊥} invariant (here we are using the finite dimensionality). Thus ρ_{ξ} ≡ ρ_{ξ}|_{W} ⊕ ρ_{ξ}|_{W^{⊥}}, contradicting irreducibility.

This has the following important consequence:

Lemma 6.2 (Schur’s lemma). If ξ, ξ′ ∈ Γ and ϕ : ρ_{ξ} → ρ_{ξ′} is an morphism, then either ϕ = 0, or ϕ is an isomorphism. If ξ = ξ′, then ϕ is a constant multiple of the identity I_{V_{ξ}}.

Proof Observe that ϕ(V_{ξ}) is an invariant subspace of V_{ξ′}, and hence is either equal to V_{ξ′} or {0}. Similarly, ker(ϕ) is an invariant subspace of V_{ξ}, and is hence equal to either V_{ξ} or {0}. Considering all the possibilities we see that ϕ is either 0 or an isomorphism.

Now suppose ξ = ξ′. Then by the first part of Schur’s lemma, we know that ϕ − λI_{V_{ξ}} (which is also a morphism) is either an isomorphism or zero for every λ ∈ C. Since ϕ must have at least one eigenvalue, the claim follows.

As a consequence, we can obtain the following ergodic theorem.

Lemma 6.3. Let ξ, ξ′ ∈ Γ, and let T ∈ HS(V_{ξ} → V_{ξ′}). Let ⟨T ⟩ ∈ HS(V_{ξ} → V_{ξ′}) denote the averaged operator ⟨T ⟩ := ∫_{G} ρ_{ξ′}(x)T ρ_{ξ}(x)^{−1} dx.

• If ξ ≠ ξ′, then

⟨T ⟩ = 0.  (13)
• If ξ = ξ′, then
⟨T ⟩ = \frac{tr(T)}{dim(V_{ξ})}I_{V_{ξ}}.  (14)

Proof By a change of variables z := y ∘ x we observe the morphism property

ρ_{ξ′}(y)⟨T ⟩ = ∫_{G} ρ_{ξ′}(y ∘ x)T ρ_{ξ}(x)^{−1} dx
= ∫_{G} ρ_{ξ′}(z)T ρ_{ξ}(y^{−1} ∘ z)^{−1} dz
= ⟨T ⟩ρ_{ξ}(y).

Thus by Schur’s lemma, ⟨T ⟩ is either 0 or an isomorphism. If ξ ≠ ξ′ then it cannot be an isomorphism, hence it must be 0. If ξ = ξ′ then ⟨T ⟩ is a constant multiple of the identity by Schur’s lemma. But since

tr(⟨T ⟩) = ∫_{G} tr(ρ_{ξ′}(x)T ρ_{ξ}(x)^{−1}) dx
= ∫_{G} tr(T) dx
= tr(T)

we must have tr(⟨T ⟩) = \frac{tr(T)}{dim(V_{ξ})}I_{V_{ξ}} as claimed.

7. Surjectivity and multiplicity

We now settle the surjectivity and multiplicity questions.

Proposition 7.1. The Fourier transform f ↦ f̂ is a surjection from L^{2}(G) to ∫_{Γ} HS(V_{ξ}) dξ, and m_{ξ} = dim(V_{ξ}).

Proof We define the modified measure d̃ξ by

∫_{Γ} f(ξ) d̃ξ := ∑_{ξ∈Γ} f(ξ) dim(V_{ξ});

our second task is to show that dξ = d̃ξ.

Let F be any element of ∫_{Γ} HS(V_{ξ}) dξ, so that each F (ξ) is an element of HS(V_{ξ}), and let f ∈ L^{2}(G) be the modified inverse Fourier transform

f(x) := ∫_{Γ}⟨F (ξ), ρ_{ξ}(x)⟩ d̃ξ.

We claim that F = f̂; this will prove surjectivity, and by comparing the above formula against the Fourier inversion formula (11) we see that dξ = d̃ξ (recall that F is arbitrary).

Now we show F = f̂. Expanding this, we see that we have to show that

F (ξ′) = ∫_{G} ∫_{Γ}⟨F (ξ), ρ_{ξ}(x)⟩ρ_{ξ′}(x) d̃ξ dx

for all ξ′ ∈ G. By Fubini’s theorem, it will suffice to show that

∫_{G}⟨F (ξ), ρ_{ξ}(x)⟩ρ_{ξ′}(x) dξ dx = \{ \begin{matrix}\frac{1}{dim(V_{ξ})}F (ξ) & if ξ = ξ′ \\ 0 & if ξ ≠ ξ′. \end{matrix}  (15)

To do this, we use Lemma 6.3, but with T : V_{ξ} → V_{ξ′} specialized to a rank 1 operator of the form

T_{a,a′}v := ⟨v, a⟩a′ for all v ∈ V_{ξ}

for some a ∈ V_{ξ}, a′ ∈ V_{ξ′} to be chosen later. By definition, we have

⟨T_{a,a′}⟩v = ∫_{G} ρ_{ξ′}(x)a′⟨ρ_{ξ}(x)^{−1}v, a⟩ dx.

Specializing to the case v := F (ξ)a, and then summing a over an orthonormal basis, we obtain

∑_{a}⟨T_{a,a′}⟩F (ξ)a = ∫_{G} tr(ρ_{ξ}(x)^{−1}F (ξ))ρ_{ξ′}(x)a′ dx.

When ξ ≠ ξ′, we thus see from (13) that

∫_{G} tr(ρ_{ξ}(x)^{−1}F (ξ))ρ_{ξ′}(x)a′ = 0

for all a′ ∈ V_{ξ′}, which proves the second half of (15). When ξ = ξ′, we see from (14) that

∫_{G} tr(ρ_{ξ}(x)^{−1}F (ξ))ρ_{ξ}(x)a′ dx = ∑_{a} \frac{tr(T_{a,a′})}{dim(V_{ξ})}F (ξ)a.

But tr(T_{a,a′}) = ⟨a′, a⟩, so we can rewrite the right-hand side as

\frac{1}{dim(V_{ξ})}F (ξ) ∑_{a}⟨a′, a⟩a.

Since a ranges over an orthonormal basis, ∑_{a}⟨a′, a⟩a = a′, and this proves the first half of (15).

One particular consequence of the above is

Theorem 7.2 (Peter-Weyl theorem). The convolution algebra L^{2}(G) is isomorphic to ∫_{Γ} HS(V_{ξ}) dξ, as an algebra, as a Hilbert space, and as a representation of G (this last fact follows from (12); in fact the two spaces are isomorphic for G acting either on the left or on the right).

As a particular consequence of this isomorphism we see from dimension count that

|G| = ∑_{ξ∈Γ} dim(V_{ξ})^{2};  (16)

in particular, we see that Γ is finite.

The space ∫_{Γ} HS(V_{ξ}) dξ can be thought of as a space of block-diagonal matrices, with each block having dim(V_{ξ}) rows and columns (and having weight m_{ξ} = dim(V_{ξ}). This realizes our earlier stated goal of representing the convolution operators in HS(L^{2}(G)) as block-diagonal matrices.

8. Conjugation-invariant functions

The group G acts on L^{2}(G) by conjugation, f(x) → f(y ∘ x ∘ y^{−1}), or in other words f ↦ τ_{y}fτ_{y}^{−1}. The group G similarly acts on HS(V_{ξ}) by conjugation, F (ξ) ↦ ρ_{ξ}(y)F (ξ)ρ_{ξ}(y)^{−1}, and hence acts on ∫_{Γ} HS(V_{ξ}) dξ as well. From the Peter-Weyl theorem we know that these actions are intertwined by the Fourier transform.

In particular, a function f is invariant under conjugation if and only if its Fourier transform f̂(ξ) is invariant under conjugation as well. But by Schur’s lemma (or (14)) we see that an operator in HS(V_{ξ}) is invariant under conjugation by the ρ_{ξ}(y) if and only if it is a constant multiple of the identity I_{V_{ξ}}. Thus the Fourier transform can be restricted to an isomorphism between the conjugation invariant functions L^{2}(G)^{G} of L^{2}(G), and the space ∫_{Γ} CI_{V_{ξ}} dξ.

Conjugation-invariant functions in L^{2}(G)^{G} are sometimes called class functions since they are constant on each conjugacy class. The Fourier analysis of class functions is simpler than that of general functions, because each block HS(V_{ξ}) in the matrix representation has been replaced by a scalar. To compute more effectively we introduce the notion of a character.

For each ξ ∈ Γ, define the Dirac delta mass δ_{ξ} on Γ by setting δ_{ξ}(ξ′) := 0 if ξ′ ≠ ξ and δ_{ξ}(ξ) := \frac{1}{dim(V_{ξ})}I_{V_{ξ}}. Observe that the Dirac delta masses δ_{ξ} form an orthonormal basis of ∫_{Γ} CI_{V_{ξ}} dξ.

The inverse Fourier transform of δ_{ξ} is denoted \overline{χ_{ξ}}, the conjugate irreducible character corresponding to ξ; we have

\overline{χ_{ξ}}(x) = ∫_{Γ}⟨δ_{ξ}, ρ_{ξ′}(x)⟩ dξ′

and hence

χ_{ξ}(x) = tr(ρ_{ξ}(x)).

Since the δ_{ξ} are an orthonormal basis of ∫_{Γ} CI_{V_{ξ}} dξ, the characters χ_{ξ} form an orthonormal basis of L^{2}(G)^{G}; in particular we have

f = ∫_{Γ}⟨f, χ_{ξ}⟩χ_{ξ} dξ

for all f ∈ L^{2}(G)^{G}; conjugating, we obtain

f = ∫_{Γ}⟨f, \overline{χ_{ξ}}⟩\overline{χ_{ξ}} dξ

Taking Fourier transforms, we obtain

f̂ = ∫_{Γ}⟨f, \overline{χ_{ξ}}⟩δ_{ξ} dξ

and in particular

f̂(ξ) = \frac{⟨f, \overline{χ_{ξ}}⟩}{dim(V_{ξ})}I_{V_{ξ}}.

We can in fact define a character χ_{ρ} ∈ L^{2}(G) for every representation ρ, by the same formula χ_{ρ}(x) := tr(ρ(x)). From (5) we have χ_{ρ_{1}⊕ρ_{2}} = χ_{ρ_{1}} + χ_{ρ_{2}}, so every character is a linear combination of the irreducible characters. Also, from the linear independence of the χ_{ξ} we thus see that the multiplicity of the irreducible representation ρ_{ξ} in any other representation ρ is well-defined; in fact, by orthonormality of the characters, this multiplicity is equal to ⟨χ_{ξ}, χ_{ρ}⟩. Thus one can identify the space of representations of G (modulo equivalence) with the positive integer combinations of the characters χ_{ξ}.

We now specialize to the case where G is abelian. In this case L^{2}(G)^{G} = L^{2}(G), which implies that HS(V_{ξ}) = CI_{V_{ξ}} for all ξ ∈ Γ; in other words, all irreducible representations ρ_{ξ} must be one-dimensional. Thus we may as well set V_{ξ} := C, and the characters χ_{ξ} are essentially the same as the representations ρ_{ξ}. In particular, since ρ_{ξ} is unitary, we have |χ_{ξ}(x)| = 1 for all x ∈ G.

Now observe that if ρ_{ξ} and ρ_{ξ′} are one-dimensional representations, then ρ_{ξ}ρ_{ξ′} is another one-dimensional representation, which must of course be irreducible. We can thus define an addition operation on Γ by setting ρ_{ξ+ξ′} := ρ_{ξ}ρ_{ξ′}. This can easily be verified to turn Γ into an abelian group, sometimes called the dual group G^{∗} of G; from (16) we thus see that |Γ| = |G|. This theory then collapses to the familiar theory of the Fourier transform on abelian groups.

The trick of multiplying two representations ρ_{ξ} and ρ_{ξ′} to form a third representation ρ_{ξ}ρ_{ξ′} is not restricted to one-dimensional representations; it is easy to take two higher-dimensional representations ρ and ρ′ and form a tensor product ρ ⊗ ρ′ of the two representations. On the character level, this corresponds to pointwise multiplication of the two characters. The new feature in the non-abelian case, however, is that the tensor product of two irreducible representations need not be irreducible. An equivalent formulation of this is that the product of two irreducible characters is character, but not necessarily irreducible; it may instead split as a nonnegative integer combination of irreducible characters. The structure constants of this multiplication rule are extremely rich combinatorially, and are important in a vast number of applications, but will not be discussed here.

Department of Mathematics, UCLA, Los Angeles, CA 90024

   E-mail address: tao@math.ucla.edu