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.
At first sight this looks like a step backward. Much of a mathematical education
runs toward abstraction. One starts with matrices and concrete vectors, then learns
to see them as instances of abstract vector spaces and linear maps. A group is
eventually freed from any particular matrix realization and studied as an abstract
object in its own right. Representation theory appears to reverse that motion,
sending an abstract group back into matrices.
The resolution is that the two directions serve different ends. Abstraction isolates
what a group is, independent of how it happens to be presented.
Representation theory is how one computes with it. A group on its own
offers no numbers to calculate, no eigenvalues, no trace, no characteristic
polynomial, whereas a linear map on a vector space offers all of them. To represent
a group is to send it into the one setting where the full machinery of linear
algebra applies. Questions about the group become questions about matrices, which
the tools of the earlier pages can answer. When the representation is faithful the
group is recovered exactly from its image, so that no information is lost in the
passage. A single well-chosen representation can already capture the whole group.
The abstraction was never abandoned. It is what tells us which matrix realizations
are the same group wearing different clothes.
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 such as a temperature or a mass
density, the rotation does nothing at all. The number is unchanged. On a vector
quantity such as a velocity or a force, the rotation acts by the usual
\(3 \times 3\) rotation matrix. On a quantity built from products of vectors, such
as a stress tensor or 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. Representation theory makes this precise. It catalogues the
distinct ways a single group can act linearly.
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 that makes this organization possible:
representation, irreducibility, and the intertwining map. The later pages on tensor
products and compact groups build upon it.
Representations
We met representations once before, in the abstract setting of a Lie group acting
smoothly on a vector space. There 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 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 of the compact groups in the pages ahead. 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 below.
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. Exhibiting a group as matrices
is not the aim of the theory, however, since the groups we study are already matrix
groups. The aim is to determine, up to isomorphism, all the ways a fixed
group can act linearly. The alternative notation \(g \cdot v\) for \(\Pi(g)v\) was
introduced for group actions, and this catalogue is the precise sense in which that
shorthand can stand for different linear maps on different representation spaces.
Each representation is one such way.
Irreducibility
Given 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. 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 rather than 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.
It is 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 is to
determine, up to isomorphism, all the irreducible representations of a given group,
and we take it up for \(\mathfrak{sl}(2; \mathbb{C})\) and for the compact groups in
the pages ahead. 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, such as atoms in a molecule or 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. The single shorthand \(g \cdot x\) then
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 is more delicate: whether
every representation \(\pi\) of \(\mathfrak{g}\) arises from a representation
\(\Pi\) of \(G\). Such a lift exists when \(G\) is simply connected and fails to
exist 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
\(\pi(X) = \frac{d}{dt}\Pi(e^{tX})|_{t=0}\). Irreducibility of \(\pi\) then
forces \(W = \{0\}\) or \(W = V\), so \(\Pi\) 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 is possible exactly on the
identity component. Near the identity the exponential map is a local diffeomorphism,
so its image contains a neighborhood of \(I\), and such a
neighborhood generates all of a connected group.
Granting that hypothesis, the study of a connected group's representations reduces
to the linear-algebraic study of its Lie algebra's representations. That strategy
underlies 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
\(\mathfrak{g}_{\mathbb{C}}\). That is 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, so irreducibility is unaffected
by the passage to \(\mathfrak{g}_{\mathbb{C}}\). A complex subspace \(W\) is
invariant under \(\pi(X + iY)\) exactly when it is invariant under both \(\pi(X)\)
and \(\pi(Y)\). 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.