Introduction
When a Lie group acts on a space, we have so far asked what the action does to
points: which points move where, which stay fixed, how orbits foliate the space.
But many of the spaces that matter in applications are not just sets of points; they are
vector spaces, and the group acts on them
linearly.
A rotation does not merely permute the vectors of \(\mathbb{R}^3\); it acts on them by a
linear map, and that map can be written as a matrix. The study of how a group acts by
linear maps on vector spaces is representation theory, and it is the
subject of this page.
There is a subtlety hidden in the familiar shorthand \(g \cdot x\) for a group action.
Consider a single rotation \(R \in SO(3)\) acting on different kinds of data attached to a
point in space. On a scalar quantity — a temperature, a mass density — the rotation does
nothing at all: the number is unchanged. On a vector quantity — a velocity, a force — the
rotation acts by the usual \(3 \times 3\) rotation matrix. On a quantity built from
products of vectors — a stress tensor, a moment of inertia — it acts by a more elaborate
rule. The same group element \(R\) thus produces genuinely different linear maps
depending on the type of object it acts on, even though we write \(g \cdot x\) the same way
in each case. Making this precise — cataloguing the distinct ways a single group can act
linearly — is exactly what representation theory provides.
This question is not merely organizational. In the
discrete symmetry
of a regular polygon, the symmetry group \(D_n\) acts on the plane and preserves the shape;
passing to the continuous limit gave us the rotation groups \(SO(2)\) and \(SO(3)\) as
matrix Lie groups. Now we ask how those continuous groups act not on a single fixed object
but on the various feature spaces that data can inhabit. The answer organizes those feature
spaces into irreducible building blocks, and the way a group acts on each block is the
information a symmetry-aware computation must respect. We will develop the vocabulary —
representation, irreducibility, intertwining map — that makes this organization possible,
and that the later pages on tensor products and compact groups will build upon.
Representations
We met representations once before, in the abstract setting of a Lie group acting
smoothly on a vector space: a
representation
was a Lie group homomorphism \(\rho : G \to GL(V)\), and a smooth linear action turned
out to be exactly such a homomorphism. Here we specialize to the setting of this section
of the curriculum — matrix Lie groups — and adopt the concrete viewpoint
that will carry us through irreducibility, tensor products, and the representation theory
of compact groups. The abstract definition and the matrix definition agree wherever both
apply; we keep them as two layers of one idea, writing \(\rho\) for the abstract action
and \(\Pi\) for its matrix-group incarnation.
Throughout, if \(V\) is a finite-dimensional real or complex vector space, \(GL(V)\)
denotes the group of invertible linear transformations of \(V\). A choice of basis
identifies \(GL(V)\) with \(GL(n, \mathbb{R})\) or \(GL(n, \mathbb{C})\), and the
resulting topology on \(GL(V)\) does not depend on that choice; we regard \(GL(V)\) as a
matrix Lie group. Similarly, \(\mathfrak{gl}(V) = \operatorname{End}(V)\) denotes the
space of all linear operators from \(V\) to itself, a Lie algebra under the commutator
bracket \([X, Y] = XY - YX\).
Definition: Representation of a Matrix Lie Group
Let \(G\) be a
matrix Lie group.
A finite-dimensional complex representation of \(G\) is a Lie group
homomorphism
\[
\Pi : G \to GL(V),
\]
where \(V\) is a finite-dimensional complex vector space with \(\dim(V) \geq 1\). A
finite-dimensional real representation of \(G\) is defined in the same
way, with \(V\) a finite-dimensional real vector space. The space \(V\) is the
representation space, and we say \(\Pi\) is a representation of \(G\)
acting on \(V\).
Unless we say otherwise, "representation" will mean finite-dimensional complex
representation: the complex case is where the structure theory is cleanest, and it is the
setting in which we will classify the irreducible representations of \(\mathfrak{sl}(2;
\mathbb{C})\) and the compact groups later in the track. The real case is recovered as a
special situation whenever a complex representation happens to preserve a real subspace.
The same definition has a Lie-algebra counterpart, obtained by replacing the group
homomorphism with a
Lie algebra homomorphism.
This is the form in which representations of the
Lie algebra
of \(G\) will appear, and the two are linked by the exponential map in a way we make
precise on the companion page.
Definition: Representation of a Lie Algebra
Let \(\mathfrak{g}\) be a real or complex Lie algebra. A
finite-dimensional complex representation of \(\mathfrak{g}\) is a
Lie algebra homomorphism
\[
\pi : \mathfrak{g} \to \mathfrak{gl}(V),
\]
where \(V\) is a finite-dimensional complex vector space; that is, \(\pi\) is linear
and satisfies \(\pi([X, Y]) = \pi(X)\pi(Y) - \pi(Y)\pi(X)\) for all
\(X, Y \in \mathfrak{g}\). A finite-dimensional real representation
of a real Lie algebra \(\mathfrak{g}\) is defined in the same way, with \(V\) a
finite-dimensional real vector space.
A representation is called faithful if \(\Pi\) (or \(\pi\)) is one-to-one.
A faithful representation of a matrix Lie group \(G\) lets us recover \(G\), up to
isomorphism, as the group of matrices \(\{\Pi(A) : A \in G\}\); this is the historical
origin of the word "representation," since \(\Pi\) lets us represent the group as
a group of matrices. The aim of the theory, however, is not merely to exhibit a group as
matrices — the groups we study are already matrix groups — but to determine, up to
isomorphism, all the ways a fixed group can act linearly. Following the alternative
notation \(g \cdot v\) for \(\Pi(g)v\) introduced for group actions, this is the precise
sense in which the shorthand \(g \cdot v\) can stand for different linear maps on different
representation spaces: each representation is one such way.
Irreducibility
Having a representation \(\Pi\) of \(G\) on \(V\), the first structural question is whether
\(V\) can be broken into smaller pieces that the group respects. A piece the group respects
is a subspace that \(\Pi\) never maps outside of itself.
Definition: Invariant Subspace and Irreducibility
Let \(\Pi\) be a finite-dimensional representation of a matrix Lie group \(G\) acting
on a space \(V\). A subspace \(W \subseteq V\) is invariant if
\[
\Pi(A) w \in W \quad \text{for all } w \in W \text{ and all } A \in G.
\]
An invariant subspace \(W\) is nontrivial if \(W \neq \{0\}\) and
\(W \neq V\). A representation with no nontrivial invariant subspaces is called
irreducible. The terms are defined in the same way for a
representation \(\pi\) of a Lie algebra \(\mathfrak{g}\), with \(\pi(X)\) in place of
\(\Pi(A)\) and the requirement holding for all \(X \in \mathfrak{g}\).
The two trivial invariant subspaces \(\{0\}\) and \(V\) are present for every
representation, which is why they are excluded: irreducibility asks that there be
nothing else. An irreducible representation is thus an atom of the theory — a
feature type that cannot be split into smaller group-respecting pieces — and the
classification problem for a group is, in essence, the problem of finding all of its
irreducible representations.
One point of care attends the complex setting we have adopted as primary. When \(\Pi\) is
a complex representation, \(V\) is a complex vector space, and an invariant subspace
\(W\) is required to be a complex subspace — closed under multiplication
by complex scalars, not merely real ones. The same applies to complex representations of
a real Lie algebra: even though the algebra is real, the invariant subspaces relevant to
irreducibility live in the complex space \(V\) and must be complex subspaces of it. This
is not a technicality to be waved away; whether a representation is irreducible can depend
on whether one allows real or complex subspaces, and we will always mean complex unless
the real case is named explicitly.
Intertwining Maps
Irreducibility tells us when a single representation cannot be broken apart. To compare
two representations, we need a notion of a map that respects both group actions at once.
Such a map is the representation-theoretic analogue of the
equivariant map
between spaces carrying a group action: there, a map commuted with two actions on
manifolds; here, the map is linear and commutes with two linear actions.
Definition: Intertwining Map
Let \(G\) be a matrix Lie group, let \(\Pi\) be a representation of \(G\) acting on a
space \(V\), and let \(\Sigma\) be a representation of \(G\) acting on a space \(W\). A
linear map \(\phi : V \to W\) is an intertwining map of
representations if
\[
\phi(\Pi(A) v) = \Sigma(A) \phi(v) \quad \text{for all } A \in G
\text{ and all } v \in V.
\]
If \(\phi\) is in addition invertible, it is an isomorphism of
representations, and \(\Pi\) and \(\Sigma\) are said to be isomorphic.
The analogous property, with \(\pi(X)\) and \(\sigma(X)\) for \(X \in \mathfrak{g}\),
defines intertwining maps of representations of a Lie algebra.
Written in the action notation \(g \cdot v\), the defining identity reads
\(\phi(A \cdot v) = A \cdot \phi(v)\): the map \(\phi\) commutes with the action
of \(G\). This is precisely the equivariance condition from the theory of group
actions, now demanded of a linear map between two representation spaces. An
intertwining map is, in a word, a linear equivariant map — the morphism that makes
representations into a category, and the object through which the comparison "are these
two representations the same?" is posed and answered.
Two representations being isomorphic means there is an invertible linear dictionary
translating one action into the other; isomorphic representations are the same
representation in different coordinates. A central problem of the theory — the one we take
up for \(\mathfrak{sl}(2; \mathbb{C})\) and for the compact groups in the pages ahead — is
to determine, up to isomorphism, all the irreducible representations of a given group, and
intertwining maps are the tool that gives the phrase "up to isomorphism" its meaning.
Why Equivariant Networks Are Built From Representations
The vocabulary of this page is exactly the vocabulary used to design neural networks
that respect a continuous symmetry. When a network processes data living in
three-dimensional space — atoms in a molecule, points in a scene — one asks that
rotating the input rotate the output correspondingly. A layer of such a network is a
map between feature spaces, and the requirement that it commute with the rotation
action on input and output is the intertwining condition stated above. The feature
spaces themselves are organized into irreducible representations of the rotation
group — the scalars, vectors, and higher tensors with which we opened — so that the
single shorthand \(g \cdot x\) resolves into a definite linear action on each feature
type. The classification of irreducible representations and the maps between them,
developed abstractly here, is the design space of symmetry-respecting architectures.
Representations of the Group and of Its Lie Algebra
A representation of a matrix Lie group \(G\) and a representation of its Lie algebra
\(\mathfrak{g}\) are not independent objects. The
exponential map
that ties \(G\) to \(\mathfrak{g}\) carries one to the other, in the same way it carries
the group law to the bracket. The following proposition makes the link precise: every
group representation differentiates to an algebra representation, related by the
exponential.
Proposition (Differentiating a Representation)
Let \(G\) be a matrix Lie group with Lie algebra \(\mathfrak{g}\), and let \(\Pi\) be a
finite-dimensional representation of \(G\) acting on a space \(V\). Then there is a
unique representation \(\pi\) of \(\mathfrak{g}\) acting on the same space such that
\[
\Pi(e^X) = e^{\pi(X)} \quad \text{for all } X \in \mathfrak{g}.
\]
The representation \(\pi\) is given by
\[
\pi(X) = \left. \frac{d}{dt} \Pi(e^{tX}) \right|_{t=0},
\]
and satisfies \(\pi(AXA^{-1}) = \Pi(A)\pi(X)\Pi(A)^{-1}\) for all \(X \in \mathfrak{g}\)
and all \(A \in G\).
We call \(\pi\) the representation of \(\mathfrak{g}\) associated to
\(\Pi\). It is the differential of \(\Pi\) at the identity, and the relation
\(\Pi(e^X) = e^{\pi(X)}\) is the representation-level shadow of the
Lie algebra homomorphism
induced by a Lie group homomorphism. The converse question — whether every representation
\(\pi\) of \(\mathfrak{g}\) arises from a representation \(\Pi\) of \(G\) — is more
delicate; it holds when \(G\) is simply connected, and fails in general, a phenomenon tied
to the fundamental group of \(G\).
When \(G\) is connected, the passage from \(\Pi\) to \(\pi\) loses no structural
information: the two see exactly the same invariant subspaces, hence the same
irreducibility, and the same isomorphisms.
Proposition (Group–Algebra Correspondence)
Let \(G\) be a connected matrix Lie group with Lie algebra \(\mathfrak{g}\).
-
Let \(\Pi\) be a representation of \(G\) and \(\pi\) the associated representation
of \(\mathfrak{g}\). Then \(\Pi\) is irreducible if and only if \(\pi\) is
irreducible.
-
Let \(\Pi_1\) and \(\Pi_2\) be representations of \(G\), with associated algebra
representations \(\pi_1\) and \(\pi_2\). Then \(\pi_1\) and \(\pi_2\) are
isomorphic if and only if \(\Pi_1\) and \(\Pi_2\) are isomorphic.
Proof (Part 1)
Suppose \(\Pi\) is irreducible, and let \(W\) be a subspace invariant under
\(\pi(X)\) for all \(X \in \mathfrak{g}\). For \(A \in G\), connectedness lets us
write \(A = e^{X_1} \cdots e^{X_m}\) for some \(X_1, \dots, X_m \in \mathfrak{g}\).
Since \(W\) is invariant under each \(\pi(X_j)\), it is invariant under
\(\exp(\pi(X_j)) = I + \pi(X_j) + \pi(X_j)^2/2 + \cdots\), and hence under
\[
\Pi(A) = \Pi(e^{X_1}) \cdots \Pi(e^{X_m})
= e^{\pi(X_1)} \cdots e^{\pi(X_m)}.
\]
As \(\Pi\) is irreducible and \(W\) is invariant under every \(\Pi(A)\), we conclude
\(W = \{0\}\) or \(W = V\); thus \(\pi\) is irreducible. Conversely, if \(\pi\) is
irreducible and \(W\) is invariant under \(\Pi\), then \(W\) is invariant under
\(\Pi(e^{tX})\) for all \(X\), hence under \(\pi(X) = \frac{d}{dt}\Pi(e^{tX})|_{t=0}\);
so \(W = \{0\}\) or \(W = V\), and \(\Pi\) is irreducible. Part 2 follows by the same
connectedness argument applied to an intertwining map.
The hypothesis that \(G\) be connected is essential: the argument turns on writing every
group element as a product of exponentials, which
holds precisely on the identity component.
With it, the study of a connected group's representations reduces to the linear-algebraic
study of its Lie algebra's representations — the strategy underlying nearly every explicit
classification, including the one for \(\mathfrak{sl}(2; \mathbb{C})\) we take up next.
Complexification and Unitarity
Two further facts complete the basic dictionary, both reflecting the primacy of the
complex case. The first says that working with complex representations of a real Lie
algebra is the same as working with complex-linear representations of its
complexification
\(\mathfrak{g}_{\mathbb{C}}\) — the complex Lie algebra obtained by allowing complex
coefficients, in which a real algebra such as \(\mathfrak{su}(2)\) becomes
\(\mathfrak{sl}(2; \mathbb{C})\).
Proposition (Extension to the Complexification)
Let \(\mathfrak{g}\) be a real Lie algebra and \(\mathfrak{g}_{\mathbb{C}}\) its
complexification. Then every finite-dimensional complex representation \(\pi\) of
\(\mathfrak{g}\) has a unique extension to a complex-linear representation of
\(\mathfrak{g}_{\mathbb{C}}\), also denoted \(\pi\), given by
\[
\pi(X + iY) = \pi(X) + i\,\pi(Y) \quad \text{for } X, Y \in \mathfrak{g}.
\]
Moreover, \(\pi\) is irreducible as a representation of \(\mathfrak{g}_{\mathbb{C}}\)
if and only if it is irreducible as a representation of \(\mathfrak{g}\).
The two have precisely the same invariant subspaces — a complex subspace \(W\) is
invariant under \(\pi(X + iY)\) exactly when it is invariant under both \(\pi(X)\) and
\(\pi(Y)\) — so irreducibility is unaffected by the passage to
\(\mathfrak{g}_{\mathbb{C}}\). This is what licenses replacing a real algebra such as
\(\mathfrak{su}(2)\) by its
complexification \(\mathfrak{sl}(2; \mathbb{C})\)
when classifying representations, a substitution we will rely on directly.
The second fact concerns representations that preserve an inner product. On a
finite-dimensional inner product space, a unitary operator is one preserving the inner
product; a representation built from such operators is the symmetry-respecting analogue of
an orthogonal transformation.
Definition: Unitary Representation
Let \(V\) be a finite-dimensional inner product space and \(G\) a matrix Lie group. A
representation \(\Pi : G \to GL(V)\) is unitary if \(\Pi(A)\) is a
unitary operator on \(V\) for every \(A \in G\).
Proposition (Unitary Representations and Skew-Adjointness)
Let \(G\) be a matrix Lie group with Lie algebra \(\mathfrak{g}\), let \(V\) be a
finite-dimensional inner product space, let \(\Pi\) be a representation of \(G\) acting
on \(V\), and let \(\pi\) be the associated representation of \(\mathfrak{g}\). If
\(\Pi\) is unitary, then \(\pi(X)\) is skew self-adjoint — that is,
\(\pi(X)^* = -\pi(X)\) — for all \(X \in \mathfrak{g}\). Conversely, if \(G\) is
connected and \(\pi(X)\) is skew self-adjoint for all \(X \in \mathfrak{g}\), then
\(\Pi\) is unitary.
Proof (Sketch)
If \(\Pi\) is unitary, then for all \(X \in \mathfrak{g}\) and \(t \in \mathbb{R}\),
\[
(e^{t\pi(X)})^* = \Pi(e^{tX})^* = \Pi(e^{tX})^{-1} = e^{-t\pi(X)}.
\]
Differentiating at \(t = 0\) gives \(\pi(X)^* = -\pi(X)\). Conversely, if
\(\pi(X)^* = -\pi(X)\), the same computation shows each \(\Pi(e^{tX}) = e^{t\pi(X)}\)
is unitary; when \(G\) is connected, every element is a product of such exponentials,
so \(\Pi(A)\) is unitary for all \(A \in G\).
These two results round out the language of the theory: representations can be transported
between a real algebra and its complexification without disturbing irreducibility, and the
representations that preserve an inner product are exactly those whose algebra operators
are skew self-adjoint. The vocabulary is now complete — representations, irreducibility,
intertwining maps, the group–algebra correspondence, and the passage to the
complexification.