From Fourier to Compact Groups
Fourier analysis on the circle rests on a single structural fact. The exponentials
\(e^{in\theta}\) form an
orthonormal basis
of the square-integrable functions on \(S^1\). Every periodic signal decomposes into these
elementary frequencies, and the decomposition is orthogonal, complete, and norm-preserving. The
exponentials are not an arbitrary choice. The circle is a group under addition of angles, and each
\(e^{in\theta}\) is a one-dimensional representation of that group, since the map
\(\theta \mapsto e^{in\theta}\) sends the group operation to multiplication of complex numbers.
Classical Fourier analysis is, in this light, the harmonic analysis of the abelian group \(S^1\),
and the basis functions are exactly its irreducible representations.
The aim of this page is to identify the analogue of this basis for a general compact group, where the group
need not be abelian and its irreducible representations need not be one-dimensional. The replacement for the
exponentials will be the matrix entries of the irreducible representations, and the replacement for
the orthonormal basis of \(L^2(S^1)\) will be a basis of \(L^2(K)\) assembled from those entries. The central
results are the orthogonality of characters and the completeness theorem of Peter and Weyl, which together
promote the circle's Fourier expansion to a noncommutative harmonic analysis valid on every compact Lie group.
Throughout, \(K\) denotes a compact Lie group and integration over \(K\) is taken against the
normalized, bi-invariant
Haar integral,
written \(\int_K f(x)\, dx\). Normalization means \(\int_K 1\, dx = 1\), and bi-invariance means
the integral is unchanged under replacing \(f(x)\) by \(f(yx)\) or \(f(xy)\) for any fixed
\(y \in K\). Both properties are available because \(K\) is compact. The
Haar volume form
exists, is unique once normalized, and inherits invariance from the left- and right-invariance of
the volume form on the group. We will use both invariances repeatedly and without further comment.
Characters and Class Functions
The objects that play the role of frequencies are built from traces of representations. We work
throughout with finite-dimensional
representations
of \(K\) on complex vector spaces. Recall that every such representation of a compact group may be
taken to be unitary, since averaging any inner product against the Haar integral produces an
invariant one.
Definition: Character of a Representation
Let \((\Pi, V)\) be a finite-dimensional representation of a compact Lie group \(K\). The
character of \(\Pi\) is the function \(\chi_\Pi : K \to \mathbb{C}\) defined by
\[
\chi_\Pi(x) = \operatorname{tr}\big(\Pi(x)\big).
\]
In particular \(\chi_\Pi(e) = \operatorname{tr}(I) = \dim V\), so the character records the dimension of
the representation at the identity.
The character discards all information about \(\Pi\) except what survives the trace, yet it
retains enough to detect the representation completely. Two features make it the right object.
First, it is unchanged under isomorphism of representations, because the trace is invariant under
conjugation. If \(\Pi' = S\Pi S^{-1}\) then
\(\operatorname{tr}(\Pi'(x)) = \operatorname{tr}(S\Pi(x)S^{-1}) = \operatorname{tr}(\Pi(x))\).
Second, and for the same reason, the character is constant on conjugacy classes of \(K\). This
second property is fundamental enough to name.
Definition: Class Function
A function \(f : K \to \mathbb{C}\) is a class function if it is constant on conjugacy
classes, that is, if
\[
f(yxy^{-1}) = f(x) \quad \text{for all } x, y \in K.
\]
Equivalently, \(f\) is invariant under the conjugation action of \(K\) on itself.
Every character is a class function. The verification is immediate from the multiplicativity of \(\Pi\) and the
cyclic invariance of the trace: for any \(x, y \in K\),
\[
\chi_\Pi(yxy^{-1}) = \operatorname{tr}\big(\Pi(y)\Pi(x)\Pi(y)^{-1}\big) = \operatorname{tr}\big(\Pi(x)\big) = \chi_\Pi(x),
\]
where the middle equality uses \(\operatorname{tr}(ABA^{-1}) = \operatorname{tr}(B)\). The class
functions form a closed subspace of \(L^2(K)\), and the orthogonality theorem will show that the
characters of the irreducible representations sit inside this subspace as an orthonormal system.
The Peter-Weyl theorem will then show that this system is complete. It is to the space of class
functions what the exponentials \(e^{in\theta}\) are to \(L^2(S^1)\). On the circle the two
pictures coincide, because every element is its own conjugacy class and so every function is a
class function. That degeneration is one we return to once the general theorem is in hand.
Orthonormality of Characters
We equip \(L^2(K)\) with the inner product
\[
\langle f, g \rangle = \int_K \overline{f(x)}\, g(x)\, dx,
\]
linear in the second argument. The orthogonality theorem states that the characters of the
irreducible representations form an orthonormal set with respect to this product. Distinct
irreducibles have orthogonal characters, and each irreducible character has unit norm. This is the
exact analogue of the relation \(\langle e^{im\theta}, e^{in\theta} \rangle = \delta_{mn}\)
underlying ordinary Fourier series, now without any assumption of commutativity.
Theorem: Orthonormality of Irreducible Characters
Let \((\Pi, V)\) and \((\Sigma, W)\) be irreducible representations of a compact Lie group \(K\). Then
\[
\langle \chi_\Pi, \chi_\Sigma \rangle = \int_K \overline{\chi_\Pi(x)}\, \chi_\Sigma(x)\, dx =
\begin{cases} 1 & \text{if } \Pi \cong \Sigma, \\\\ 0 & \text{if } \Pi \not\cong \Sigma. \end{cases}
\]
The proof rests on two preliminary facts. The first turns the Haar integral of a representation
into a concrete geometric object, an orthogonal projection. The second computes the trace of a
tensor product. After establishing these we assemble the orthogonality relation by recognizing the
integral above as the dimension of a space of intertwining maps, which Schur's lemma then
evaluates.
Averaging as a Projection
For any finite-dimensional representation \((\Pi, V)\) of \(K\), write
\[
V^K = \{\, v \in V : \Pi(x)v = v \text{ for all } x \in K \,\}
\]
for the subspace of vectors fixed by every group element. Averaging the operators \(\Pi(x)\) over the group
produces the projection onto this subspace.
Proposition: The Averaging Projection
Let \((\Pi, V)\) be a finite-dimensional representation of a compact Lie group \(K\), and define the operator
\[
P = \int_K \Pi(x)\, dx \in \operatorname{End}(V),
\]
the integral taken entrywise in any basis. Then \(P\) is the projection of \(V\) onto \(V^K\).
It maps \(V\) into \(V^K\), and \(Pv = v\) for every \(v \in V^K\). Consequently
\(\operatorname{tr}(P) = \dim V^K\).
Proof
\(P\) maps into \(V^K\).
Fix \(y \in K\) and \(v \in V\). Using the linearity of \(\Pi(y)\), the homomorphism property
\(\Pi(y)\Pi(x) = \Pi(yx)\), and the left-invariance of the Haar integral,
\[
\Pi(y)\, Pv = \Pi(y) \int_K \Pi(x)v\, dx = \int_K \Pi(yx)v\, dx = \int_K \Pi(x)v\, dx = Pv.
\]
The third equality substitutes \(x \mapsto yx\), under which \(dx\) is invariant. Since this holds for every
\(y \in K\), the vector \(Pv\) is fixed by the representation, so \(Pv \in V^K\).
\(P\) fixes \(V^K\) pointwise.
If \(v \in V^K\) then \(\Pi(x)v = v\) for all \(x\), so
\[
Pv = \int_K \Pi(x)v\, dx = \int_K v\, dx = \left( \int_K 1\, dx \right) v = v,
\]
using the normalization \(\int_K 1\, dx = 1\). The two properties together say that \(P\)
restricts to the identity on \(V^K\) and lands in \(V^K\), so \(P\) is a projection onto
\(V^K\). The trace of a projection equals the dimension of its image, giving
\(\operatorname{tr}(P) = \dim V^K\).
When \(V\) is irreducible and nontrivial, the only vector fixed by all of \(K\) is the origin, since
\(V^K\) is an invariant subspace and irreducibility forbids a proper nonzero one. In that case \(V^K = \{0\}\)
and the proposition records that \(\int_K \Pi(x)\, dx = 0\).
The Trace of a Tensor Product
The orthogonality integral pairs two representations, and the product of their traces will be read as the trace
of a single representation on the
tensor product
space. The underlying linear-algebra identity is the following, an immediate consequence of the
characteristic property of the tensor product,
which guarantees that \(A \otimes B\) is the well-defined operator acting by \((A \otimes B)(v \otimes w) = Av \otimes Bw\).
The tensor-trace identity
For operators \(A\) on \(V\) and \(B\) on \(W\), one has \(\operatorname{tr}(A \otimes B) = \operatorname{tr}(A)\,\operatorname{tr}(B)\).
To see this, choose bases \(\{v_j\}\) of \(V\) and \(\{w_l\}\) of \(W\), so that \(\{v_j \otimes w_l\}\) is a
basis of \(V \otimes W\). The matrix of \(A \otimes B\) in this basis has diagonal entries \(A_{jj} B_{ll}\),
whence
\[
\operatorname{tr}(A \otimes B) = \sum_{j, l} A_{jj} B_{ll} = \left(\sum_j A_{jj}\right)\!\left(\sum_l B_{ll}\right) = \operatorname{tr}(A)\,\operatorname{tr}(B).
\]
Assembling the Orthogonality Relation
Two further ingredients connect the conjugate character to a dual representation. Because \(K\) is compact we may
assume each \(\Pi(x)\) unitary, so \(\Pi(x)^{-1} = \Pi(x)^*\), the conjugate transpose. Taking the trace and using
\(\operatorname{tr}(M^*) = \overline{\operatorname{tr}(M)}\),
\[
\overline{\chi_\Pi(x)} = \overline{\operatorname{tr}(\Pi(x))} = \operatorname{tr}\big(\Pi(x)^*\big) = \operatorname{tr}\big(\Pi(x^{-1})\big).
\]
The transpose of an operator has the same trace as the operator itself, so writing \(\Pi^*\) for
the
dual representation,
which acts by \(\Pi^*(x) = [\Pi(x^{-1})]^\top\), we obtain
\[
\overline{\chi_\Pi(x)} = \operatorname{tr}\big([\Pi(x^{-1})]^\top\big) = \operatorname{tr}\big(\Pi^*(x)\big) = \chi_{\Pi^*}(x).
\]
The conjugate of a character is therefore the character of the dual representation. We can now prove the theorem.
Proof of Orthonormality
From the pairing to a tensor character.
Combining the conjugate-character identity with the tensor-trace identity, the integrand of
\(\langle \chi_\Pi, \chi_\Sigma \rangle\) becomes the character of the tensor product \(\Pi^* \otimes \Sigma\):
\[
\overline{\chi_\Pi(x)}\, \chi_\Sigma(x) = \operatorname{tr}\big(\Pi^*(x)\big)\, \operatorname{tr}\big(\Sigma(x)\big) = \operatorname{tr}\big((\Pi^* \otimes \Sigma)(x)\big).
\]
Integrating gives the dimension of an invariant subspace.
Integrating over \(K\) and exchanging trace with the entrywise Haar integral, then applying the averaging
projection to the representation \(\Pi^* \otimes \Sigma\) on \(V^* \otimes W\),
\[
\langle \chi_\Pi, \chi_\Sigma \rangle = \int_K \operatorname{tr}\big((\Pi^* \otimes \Sigma)(x)\big)\, dx = \operatorname{tr}\!\left( \int_K (\Pi^* \otimes \Sigma)(x)\, dx \right) = \operatorname{tr}(P) = \dim\big( (V^* \otimes W)^K \big),
\]
where \(P\) is the projection onto the fixed subspace \((V^* \otimes W)^K\).
Identifying the fixed subspace with intertwiners.
The space
\(V^* \otimes W\) is naturally identified with \(\operatorname{Hom}(V, W)\), the linear maps
from \(V\) to \(W\), and under this identification the action of \(x \in K\) sends a map \(T\)
to \(\Sigma(x)\, T\, \Pi(x)^{-1}\). A map is fixed by every \(x\) precisely when
\(\Sigma(x) T = T \Pi(x)\) for all \(x\), that is, precisely when \(T\) is an intertwining map
between \(\Pi\) and \(\Sigma\). Hence
\[
(V^* \otimes W)^K \cong \operatorname{Hom}_K(V, W),
\]
the space of intertwiners.
Schur's lemma evaluates the dimension.
By
Schur's lemma,
a nonzero intertwiner between irreducibles is an isomorphism, so \(\operatorname{Hom}_K(V, W) = 0\) when
\(\Pi \not\cong \Sigma\). When \(\Pi \cong \Sigma\), every self-intertwiner of an irreducible complex
representation is a scalar multiple of the identity, so \(\operatorname{Hom}_K(V, W)\) is one-dimensional.
Therefore
\[
\langle \chi_\Pi, \chi_\Sigma \rangle = \dim \operatorname{Hom}_K(V, W) = \begin{cases} 1 & \Pi \cong \Sigma, \\\\ 0 & \Pi \not\cong \Sigma, \end{cases}
\]
which is the assertion of the theorem.
The orthonormality relation already separates representations. Two irreducibles with the same
character are isomorphic, because \(\langle \chi_\Pi, \chi_\Sigma \rangle = 1\) forces
\(\Pi \cong \Sigma\). What remains is the deeper question of completeness. Do the characters
exhaust the class functions, leaving no direction in \(L^2(K)\) orthogonal to all of them? That is
the content of the Peter-Weyl theorem, to which we now turn.
Matrix Entries and the Peter-Weyl Theorem
Completeness cannot be proved at the level of characters alone. A character is a single function
extracted from a representation by the trace, and the trace collapses too much. The route to
completeness passes instead through the full collection of matrix coefficients of a
representation, of which the character is the special case formed by summing the diagonal. We
isolate these coefficients, prove that they are dense in \(L^2(K)\), and only then descend to
characters by averaging over conjugacy.
Definition: Matrix Entries of a Representation
Let \((\Pi, V)\) be a finite-dimensional representation of \(K\) and fix a basis \(\{v_j\}\) of \(V\). The
matrix entries of \(\Pi\) are the functions \(K \to \mathbb{C}\) given by
\[
x \longmapsto \big(\Pi(x)\big)_{jk},
\]
the \((j,k)\) entry of the matrix of \(\Pi(x)\) in the basis \(\{v_j\}\). More generally, and independently of
any basis, a function of the form
\[
f(x) = \operatorname{tr}\big(\Pi(x) A\big), \quad A \in \operatorname{End}(V),
\]
is called a matrix entry for \(\Pi\). Choosing \(A\) to be the matrix unit with a single \(1\)
in position \((k,j)\) recovers the coordinate function \((\Pi(x))_{jk}\), and a general \(A\)
produces a linear combination of coordinate functions. Taking \(A = I\) gives the character
\(\chi_\Pi\).
The basis-free description \(f(x) = \operatorname{tr}(\Pi(x)A)\) is the one we use, because it transforms cleanly
under the group operations. Of these, the single fact the completeness proof relies on is how an integral of
conjugated operators behaves on an irreducible representation.
Conjugation-averaging on an irreducible
If \((\Pi, V)\) is irreducible, then for every operator \(A\) on \(V\),
\[
\int_K \Pi(y)\, A\, \Pi(y)^{-1}\, dy = c\, I \quad \text{for some scalar } c,
\]
and taking traces on both sides fixes the scalar as \(c = \operatorname{tr}(A)/\dim V\). Here
the integral commutes with the trace, and
\(\operatorname{tr}(\Pi(y)A\Pi(y)^{-1}) = \operatorname{tr}(A)\) is constant in \(y\). To see
that the integral is a scalar, let \(B\) denote the operator on the left. For any \(x \in K\),
the left-invariance of the Haar integral gives
\[
\Pi(x)\, B\, \Pi(x)^{-1} = \int_K \Pi(xy)\, A\, \Pi(xy)^{-1}\, dy = \int_K \Pi(y)\, A\, \Pi(y)^{-1}\, dy = B,
\]
so \(B\) commutes with every \(\Pi(x)\). By
Schur's lemma
a self-intertwiner of an irreducible complex representation is a scalar, hence \(B = cI\).
The Algebra of Matrix Entries
Let \(\mathcal{A} \subseteq C(K)\) be the set of all functions expressible as a finite linear combination of
matrix entries, ranging over all finite-dimensional representations of \(K\). The Peter-Weyl theorem asserts that
\(\mathcal{A}\) is uniformly dense in \(C(K)\), and therefore dense in \(L^2(K)\). We prove this by verifying that
\(\mathcal{A}\) satisfies the hypotheses of the
Stone-Weierstrass theorem
on the compact Hausdorff space \(K\): that it is a self-adjoint subalgebra containing the constants and separating
points. The uniform closure of such an algebra is then all of \(C(K)\).
Theorem: Peter-Weyl (Density of Matrix Entries)
Let \(K\) be a compact matrix Lie group. The space \(\mathcal{A}\) of finite linear combinations of matrix
entries of finite-dimensional representations of \(K\) is dense in \(C(K)\) in the uniform norm, and hence
dense in \(L^2(K)\).
Proof
\(\mathcal{A}\) is an algebra.
Sums and scalar multiples of matrix entries are matrix entries by construction, so \(\mathcal{A}\) is a linear
subspace. For products, the tensor-trace identity gives, for representations \(\Pi, \Sigma\) and operators
\(A, B\),
\[
\operatorname{tr}\big(\Pi(x)A\big)\, \operatorname{tr}\big(\Sigma(x)B\big) = \operatorname{tr}\big((\Pi \otimes \Sigma)(x)\, (A \otimes B)\big),
\]
which is a matrix entry of the tensor product representation \(\Pi \otimes \Sigma\). The product of two
elements of \(\mathcal{A}\) is therefore again in \(\mathcal{A}\), so \(\mathcal{A}\) is closed under
multiplication.
\(\mathcal{A}\) contains the constants.
The trivial representation \(x \mapsto 1\) on \(\mathbb{C}\) has the constant function \(1\) as its matrix
entry, so every constant lies in \(\mathcal{A}\). In particular \(\mathcal{A}\) vanishes nowhere.
\(\mathcal{A}\) is self-adjoint.
The complex conjugate of a matrix entry is a matrix entry of the dual representation. Indeed, for unitary
\(\Pi\), conjugating \(\operatorname{tr}(\Pi(x)A)\) and using \(\overline{\operatorname{tr}(M)} = \operatorname{tr}(M^*)\) together with the
identification of the conjugate representation with the dual \(\Pi^*\) expresses \(\overline{f}\) as a matrix
entry of \(\Pi^*\). Hence \(\mathcal{A}\) is closed under complex conjugation.
\(\mathcal{A}\) separates points.
This is the one step that uses the
hypothesis that \(K\) is a matrix Lie group. By definition such a group comes with a faithful
finite-dimensional representation \(\Pi_0\), namely the inclusion
\(K \hookrightarrow GL(n, \mathbb{C})\) itself. If \(x \neq y\) in \(K\), then
\(\Pi_0(x) \neq \Pi_0(y)\) by faithfulness, so some matrix entry \((\Pi_0)_{jk}\) takes
different values at \(x\) and \(y\). Thus \(\mathcal{A}\) separates the points of \(K\).
Conclusion via Stone-Weierstrass.
The uniform closure \(\overline{\mathcal{A}}\) is a closed subalgebra of \(C(K)\) containing the constants,
self-adjoint, and separating points. By the
Stone-Weierstrass theorem,
\(\overline{\mathcal{A}} = C(K)\). Since \(K\) is compact, \(C(K)\) is dense in \(L^2(K)\), so \(\mathcal{A}\)
is dense in \(L^2(K)\) as well.
From Matrix Entries to Characters
Density of matrix entries is the Peter-Weyl theorem in its primary form. Completeness of
characters is the statement that a continuous class function orthogonal to every irreducible
character must vanish, and it follows by projecting matrix entries onto the class functions. The
bridge is an averaging over conjugacy that sends each matrix entry to a linear combination of
characters.
Corollary: Completeness of Characters
Let \(K\) be a compact matrix Lie group. If \(f\) is a continuous class function on \(K\) satisfying
\(\langle \chi_\Pi, f \rangle = 0\) for every irreducible representation \(\Pi\), then \(f = 0\). Equivalently,
the irreducible characters form a complete orthonormal system in the Hilbert space of square-integrable class
functions.
Proof
Conjugation-averaging sends matrix entries to characters.
For a function \(g \in \mathcal{A}\), define its average over conjugacy classes,
\[
(\mathcal{C}g)(x) = \int_K g\big(y^{-1} x y\big)\, dy.
\]
The result is a class function, by the invariance of the Haar integral under \(y \mapsto zy\). Applying this
to a single matrix entry \(g(x) = \operatorname{tr}(\Pi(x) A)\) and using the homomorphism property,
\[
(\mathcal{C}g)(x) = \int_K \operatorname{tr}\big(\Pi(y^{-1}) \Pi(x) \Pi(y) A\big)\, dy = \operatorname{tr}\!\left( \Pi(x) \int_K \Pi(y)\, A\, \Pi(y)^{-1}\, dy \right),
\]
where the second equality uses the cyclic invariance of the trace and \(\Pi(y^{-1}) = \Pi(y)^{-1}\). By the
conjugation-averaging identity, the inner integral equals \(cI\) on each irreducible constituent of \(\Pi\),
so \(\mathcal{C}g\) is a linear combination of the characters \(\chi_\Pi\). Thus conjugation-averaging carries
\(\mathcal{A}\) into the span of irreducible characters.
A class function orthogonal to all characters is orthogonal to its own approximants.
Suppose \(f\) is a continuous class function with \(\langle \chi_\Pi, f \rangle = 0\) for every irreducible
\(\Pi\). By the density of \(\mathcal{A}\), choose \(g_n \in \mathcal{A}\) converging to \(f\) in \(L^2(K)\).
Each conjugation-average \(\mathcal{C}g_n\) is a linear combination of characters, so
\(\langle \mathcal{C}g_n, f \rangle = 0\). Because \(f\) is itself a class function, averaging the integrand of
\(\langle g_n, f \rangle\) over conjugacy does not change its value:
\[
\langle g_n, f \rangle = \int_K \overline{g_n(x)}\, f(x)\, dx = \int_K \overline{(\mathcal{C}g_n)(x)}\, f(x)\, dx = \langle \mathcal{C}g_n, f \rangle = 0.
\]
The middle equality holds because \(f\) is a class function. Writing the conjugation average
as a double integral and substituting \(x \mapsto y x y^{-1}\) in the inner integral returns
the original integrand after averaging over \(y\). Under that substitution \(dx\) is invariant
and \(f(y^{-1}xy) = f(x)\).
Passing to the limit.
Since \(g_n \to f\) in \(L^2(K)\), we have
\(\langle g_n, f \rangle \to \langle f, f \rangle = \|f\|_2^2\). But every
\(\langle g_n, f \rangle = 0\), so \(\|f\|_2 = 0\) and \(f = 0\) almost everywhere. Being
continuous, \(f\) is identically zero. The irreducible characters therefore leave no nonzero
class function orthogonal to all of them, which is their completeness.
The Peter-Weyl Decomposition and the Window to GDL
The density of matrix entries upgrades at once to an orthonormal basis. The space \(L^2(K)\) is a
Hilbert space.
It carries the inner product \(\langle f, g \rangle = \int_K \overline{f}\, g\, dx\) and is
complete, since \(L^2(K)\) is an instance of the
Riesz-Fischer theorem.
In a Hilbert space a dense subspace contains an orthonormal basis, and the orthogonality relations
established earlier already organize the matrix entries into orthogonal families. The combination
yields the decomposition that is the harmonic-analytic core of the theory.
Theorem: The Peter-Weyl Decomposition of \(L^2(K)\)
Let \(K\) be a compact matrix Lie group, and let \(\widehat{K}\) be a set of representatives of the
isomorphism classes of irreducible unitary representations. For each \(\Pi \in \widehat{K}\) of dimension
\(d_\Pi\), fix an orthonormal basis of its space, giving matrix entries \((\Pi(x))_{jk}\). Then the rescaled
matrix entries
\[
\sqrt{d_\Pi}\,(\Pi(x))_{jk}, \quad \Pi \in \widehat{K}, \quad 1 \leq j, k \leq d_\Pi,
\]
form an orthonormal basis of \(L^2(K)\). Equivalently, \(L^2(K)\) decomposes as the Hilbert-space direct sum
\[
L^2(K) = \bigoplus_{\Pi \in \widehat{K}} \mathcal{E}_\Pi,
\]
where \(\mathcal{E}_\Pi\) is the \(d_\Pi^2\)-dimensional space spanned by the entries of \(\Pi\).
Proof
Orthogonality and normalization.
The orthogonality relations for matrix
entries are obtained by the same mechanism that gave the character relations, applied to the
full coefficients rather than their traces. Pairing an entry of \(\Pi\) with an entry of
\(\Sigma\) and integrating produces, via the averaging projection and Schur's lemma, the
integral of a coefficient of \(\Pi^* \otimes \Sigma\) over \(K\). For inequivalent
irreducibles this vanishes, and for \(\Pi = \Sigma\) it reduces to a multiple of the identity
whose scale is fixed by taking the trace. The outcome is that entries of inequivalent
irreducibles are orthogonal in \(L^2(K)\), while within a single irreducible \(\Pi\),
\[
\big\langle (\Pi)_{jk}, (\Pi)_{lm} \big\rangle = \frac{1}{d_\Pi}\, \delta_{jl}\, \delta_{km},
\]
the factor \(1/d_\Pi\) arising because the relevant projection has trace one on a \(d_\Pi\)-dimensional space.
Rescaling each entry by \(\sqrt{d_\Pi}\) therefore produces an orthonormal family. These inner products are
finite because \(K\) is compact, so every continuous function lies in \(L^2(K)\) and the
Cauchy-Schwarz inequality
controls the pairings.
Completeness.
By the density theorem of the previous section, the span of all matrix entries is dense in \(L^2(K)\). An
orthonormal family whose span is dense is an orthonormal basis. Hence the rescaled entries form an orthonormal
basis, and grouping them by representation gives the stated orthogonal decomposition into the finite-dimensional
blocks \(\mathcal{E}_\Pi\).
Every \(f \in L^2(K)\) thus expands in matrix entries, with coefficients given by integration
against the conjugated entries. These coefficients are the noncommutative Fourier coefficients of
\(f\). The expansion converges in \(L^2(K)\) and obeys a Parseval identity summing \(|{\cdot}|^2\)
over all entries. This is the precise sense in which the Peter-Weyl theorem is the harmonic
analysis of a compact group. It furnishes the basis, the transform, and the energy identity, with
the irreducible representations playing the role of frequencies.
The Circle Recovered
When \(K = S^1\) the theory collapses onto classical Fourier series. The circle is abelian, so by
Schur's lemma every irreducible representation is one-dimensional. The irreducible representations
are exactly the characters \(\Pi_n(e^{i\theta}) = e^{in\theta}\) for \(n \in \mathbb{Z}\). Each is
its own single matrix entry, and the dimension factor \(d_\Pi = 1\) makes the rescaling trivial.
The Peter-Weyl decomposition then states that \(\{e^{in\theta}\}_{n \in \mathbb{Z}}\) is an
orthonormal basis of \(L^2(S^1)\), which is exactly the
completeness of the trigonometric system
established for Fourier series. The general theorem is the noncommutative extension of that one
fact. Where the circle offered scalar exponentials indexed by integers, a nonabelian compact group
offers matrix-valued entries indexed by its irreducible representations.
The window to geometric deep learning
The Peter-Weyl decomposition is the analytic foundation for learning architectures that
respect a continuous symmetry. When data lives on a homogeneous space of a compact group, the
natural way to process it equivariantly is to expand it in the matrix entries of the group's
irreducible representations, exactly the basis this theorem provides. Signals on the sphere
under the rotation group \(SO(3)\) are a standard instance. The coefficients in each block
\(\mathcal{E}_\Pi\) transform among themselves under the group action and never mix across
blocks, because the blocks are the irreducible constituents.
A feature indexed by a fixed irreducible is precisely a quantity that transforms in a
prescribed, irreducible way under rotation. Convolution against a function on the group
becomes, in this basis, a block-diagonal operation acting independently on each irreducible
component. The mathematical content that makes such architectures well-defined is the
decomposition proved here, namely that the irreducible blocks exhaust the function space and
are mutually orthogonal. The deeper development, in which these blocks become the typed
features of equivariant networks on homogeneous spaces, belongs to the geometric deep learning
track and is taken up there.