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: 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 \(\int_K 1\, dx = 1\); 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
\[
f(yxy^{-1}) = f(x) \qquad \text{for all } x, y \in K.
\]
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 — a degeneration 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 \(V^K\); hence \(P\) 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 — whether 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
\[
f(x) = \operatorname{tr}\big(\Pi(x) A\big), \qquad A \in \operatorname{End}(V),
\]
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 — the integral commutes with the trace, and
\(\operatorname{tr}(\Pi(y)A\Pi(y)^{-1}) = \operatorname{tr}(A)\) is constant in \(y\) — fixes the scalar as
\(c = \operatorname{tr}(A)/\dim V\). 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 \(\Pi_0\) — 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. The completeness of characters —
the statement that a continuous class function orthogonal to every irreducible character must vanish —
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 — under which \(dx\) is invariant
and \(f(y^{-1}xy) = f(x)\) — returns the original integrand after averaging over \(y\).
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 Decomposition of L²(K) 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 of the second
section 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 L²(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
\[
\sqrt{d_\Pi}\,(\Pi(x))_{jk}, \qquad \Pi \in \widehat{K}, \quad 1 \leq j, k \leq d_\Pi,
\]
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 — 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; they 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 — signals on the sphere under the
rotation group \(SO(3)\), for instance — 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. 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 — that the
irreducible blocks exhaust the function space and are mutually orthogonal — is the decomposition proved
here. 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.