Introduction
We have the vocabulary of representation theory in hand: a
representation
of a matrix Lie group is a homomorphism into the invertible linear maps of a vector
space, an
irreducible
representation is one with no nontrivial invariant subspace, and an
intertwining map
is a linear map commuting with two such actions. What the vocabulary still lacks is
examples: concrete representations to which the definitions apply and on which the
classification problem can be tested.
This page supplies those examples. We begin with three representations that every
matrix Lie group carries automatically. The standard
representation is the one by which the group is presented, the
trivial representation ignores the group entirely, and the
adjoint representation is the action of the group on its own Lie
algebra. These cost little, since the first two are immediate from the definitions and
the third we have already constructed in another guise.
The substance of the page is a single family of representations of \(SU(2)\), one for each
nonnegative integer \(m\), built on the spaces of homogeneous polynomials in two complex
variables. This family is the first place in the theory where irreducibility is not
automatic but must be proved. The proof introduces a technique that tracks how
the Lie algebra acts on eigenvectors of a single distinguished operator. That
technique organizes the representation theory of \(SU(2)\) and, beyond it, of
every compact group encountered in later pages. The distinguished operator is the
diagonalizable generator \(H\) that became available only after
complexifying
\(\mathfrak{su}(2)\) to \(\mathfrak{sl}(2; \mathbb{C})\). The present page is where that
construction earns its keep. By the end we will have, for \(SU(2)\), an explicit
irreducible representation in each dimension, together with a method for analyzing it that
generalizes.
Standard, Trivial, and Adjoint
Three representations are available for every matrix Lie group with no further work. We
record them, since each will serve as a reference point and the third closes a loop opened
on an earlier page.
The Standard Representation
A matrix Lie group \(G\) is by definition a subgroup of some \(GL(n; \mathbb{C})\), so its
elements already are invertible linear maps of \(\mathbb{C}^n\). The inclusion
map is therefore a representation without modification.
Definition: Standard Representation
Let \(G \subseteq GL(n; \mathbb{C})\) be a matrix Lie group. The
standard representation of \(G\) is the inclusion map
\[
\Pi : G \to GL(n; \mathbb{C}), \quad \Pi(A) = A,
\]
acting on \(V = \mathbb{C}^n\). If \(G\) is contained in \(GL(n; \mathbb{R})\), the
same map, acting on \(\mathbb{R}^n\), is the standard representation regarded as a
real representation. For a matrix Lie algebra \(\mathfrak{g} \subseteq
M_n(\mathbb{C})\), the standard representation is likewise
\(\pi(X) = X\).
Thus the standard representation of
\(SO(3)\)
is its usual action on \(\mathbb{R}^3\), and the standard representation of
\(SU(2)\)
is its usual action on \(\mathbb{C}^2\). These are the actions through which the groups
were first presented. Naming them as representations simply records that the presentation
is itself an example of the structure under study.
The Trivial Representation
At the opposite extreme, a group can act on a one-dimensional space by doing nothing.
Definition: Trivial Representation
Let \(G\) be a matrix Lie group. The trivial representation of \(G\)
is the map
\[
\Pi : G \to GL(1; \mathbb{C}), \quad \Pi(A) = I \quad \text{for all } A \in G,
\]
acting on \(V = \mathbb{C}\). For a Lie algebra \(\mathfrak{g}\), the
trivial representation is \(\pi : \mathfrak{g} \to
\mathfrak{gl}(1; \mathbb{C})\), \(\pi(X) = 0\) for all \(X \in \mathfrak{g}\).
The trivial representation is
irreducible.
Its representation space \(\mathbb{C}\) is one-dimensional, so its only
subspaces are \(\{0\}\) and itself, and neither is a nontrivial invariant
subspace. This representation records the scalar quantities on which a rotation
acts by the identity. It is also the building block to which every higher
representation is compared when one asks how many copies of "the trivial action"
a space contains.
The Adjoint Representation
The third automatic representation is one we have already met. In the
previous development of the Lie correspondence,
the group was shown to act on its own Lie algebra by conjugation, and the derivative of
that action recovered the bracket. Those constructions are precisely the adjoint
representations of the group and of the algebra.
The
adjoint representation of the group
is the map \(\mathrm{Ad} : G \to GL(\mathfrak{g})\), \(\mathrm{Ad}(A)(X) = AXA^{-1}\),
acting on the Lie algebra \(\mathfrak{g}\) viewed as a complex (or real) vector space. Its
derivative at the identity is the
adjoint representation of the algebra,
\(\mathrm{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\), \(\mathrm{ad}(X)(Y) =
[X, Y]\), which
is a Lie algebra homomorphism.
In the language now available, \(\mathrm{Ad}\) and \(\mathrm{ad}\) are the representations
of \(G\) and \(\mathfrak{g}\) on the vector space \(\mathfrak{g}\) itself, and the
differentiation relating them is the same passage from group to algebra representation
that holds in general. The conjugation action that once described how the group reshapes
its own infinitesimal generators is, from the present vantage, just another representation.
For \(SO(3)\) a coincidence is worth noting. The standard representation acts on
\(\mathbb{R}^3\), and the adjoint representation acts on \(\mathfrak{so}(3)\), which is
also three-dimensional and real. One can verify that the two are in fact isomorphic
as representations. This agreement is special to \(SO(3)\). It reflects the
identification of \(\mathbb{R}^3\) with \(\mathfrak{so}(3)\) that the bracket
realizes as the cross product, and it does not persist for larger groups, where
the adjoint representation has the dimension of the group rather than of the
space it was presented on.
The Representations of \(SU(2)\)
We now construct the central family of the page: an infinite list of representations of
\(SU(2)\), one for each nonnegative integer \(m\), realized on spaces of polynomials.
Unlike the three examples above, these are not handed to us by the definitions. Each
must be built and then verified to be a representation, and their irreducibility,
taken up in the next section, requires a genuine argument.
The Polynomial Spaces \(V_m\)
Fix an integer \(m \geq 0\) and let \(V_m\) be the space of homogeneous
polynomials of degree \(m\) in two complex variables \(z_1, z_2\). A typical
element is
\[
f(z_1, z_2) = a_0 z_1^m + a_1 z_1^{m-1} z_2 + a_2 z_1^{m-2} z_2^2 + \cdots
+ a_m z_2^m,
\]
with coefficients \(a_0, \dots, a_m \in \mathbb{C}\). Every term has total degree
exactly \(m\).
The monomials
\[
z_1^m,\ z_1^{m-1} z_2,\ \dots,\ z_1 z_2^{m-1},\ z_2^m
\]
form a basis. There is one such monomial for each power \(k = 0, 1, \dots, m\)
of \(z_2\), so \(\dim(V_m) = m + 1\). The small cases are familiar. The space
\(V_0 = \mathbb{C}\) consists of the constants, and \(V_1\), spanned by \(z_1\)
and \(z_2\), is two-dimensional.
The Action \(\Pi_m\)
A matrix \(U \in SU(2)\) acts on the variables \(z = (z_1, z_2) \in \mathbb{C}^2\) by the
standard representation, \(z \mapsto Uz\). To turn this into an action on
functions of \(z\), we let \(U\) act on a polynomial by substituting the
transformed variables. The substitution uses the inverse of \(U\).
Definition: The Representations \(\Pi_m\) of \(SU(2)\)
For each integer \(m \geq 0\) and each \(U \in SU(2)\), define a linear map
\(\Pi_m(U)\) on \(V_m\) by
\[
\bigl[\Pi_m(U) f\bigr](z) = f(U^{-1} z), \quad z \in \mathbb{C}^2.
\]
Because \(U^{-1}\) acts linearly on \(z\) and \(f\) is homogeneous of degree \(m\),
the composition \(f(U^{-1} z)\) is again a homogeneous polynomial of degree \(m\), so
\(\Pi_m(U)\) maps \(V_m\) into \(V_m\). The map \(\Pi_m : SU(2) \to GL(V_m)\) is a
representation
of \(SU(2)\).
The inverse is not decoration. It is what makes \(\Pi_m\) a homomorphism rather than
its reverse. Computing the composition of two such maps, we obtain
\[
\begin{align*}
\bigl[\Pi_m(U_1)\,\Pi_m(U_2) f\bigr](z)
&= \bigl[\Pi_m(U_2) f\bigr](U_1^{-1} z)
= f\bigl(U_2^{-1} U_1^{-1} z\bigr) \\\\
&= f\bigl((U_1 U_2)^{-1} z\bigr)
= \bigl[\Pi_m(U_1 U_2) f\bigr](z).
\end{align*}
\]
The identity \((U_1 U_2)^{-1} = U_2^{-1} U_1^{-1}\) reverses the order a second
time and restores \(\Pi_m(U_1)\Pi_m(U_2) = \Pi_m(U_1 U_2)\). Had we substituted
\(Uz\) instead of \(U^{-1} z\), the two reversals would not occur and the
assignment would satisfy \(\Pi_m(U_1)\Pi_m(U_2) = \Pi_m(U_2 U_1)\). That is an
anti-homomorphism, not a representation.
The Associated Lie Algebra Representation
Every group representation
differentiates to a representation of the Lie algebra.
We compute the associated representation \(\pi_m\) of \(\mathfrak{su}(2)\) directly from
the definition \(\pi_m(W) = \frac{d}{dt}\big|_{t=0} \Pi_m(e^{tW})\). For \(W \in
\mathfrak{su}(2)\) and \(f \in V_m\),
\[
\bigl(\pi_m(W) f\bigr)(z) = \left. \frac{d}{dt} \right|_{t=0}
f\bigl(e^{-tW} z\bigr).
\]
The inverse \(U^{-1} = e^{-tW}\) appears because \(\Pi_m\) was built with an inverse.
Writing \(z(t) = e^{-tW} z\), so that \(\frac{d}{dt} z(t)\big|_{t=0} = -Wz\), the
chain rule gives a first-order differential operator:
\[
\bigl(\pi_m(W) f\bigr)(z)
= -\frac{\partial f}{\partial z_1}\bigl(W_{11} z_1 + W_{12} z_2\bigr)
-\frac{\partial f}{\partial z_2}\bigl(W_{21} z_1 + W_{22} z_2\bigr).
\]
To make the action concrete it is best to pass to the
complexification \(\mathfrak{su}(2)_{\mathbb{C}} \cong \mathfrak{sl}(2; \mathbb{C})\).
By the
unique complex-linear extension,
\(\pi_m\) extends to \(\mathfrak{sl}(2; \mathbb{C})\) by the very same formula, now with
\(W\) ranging over all of \(\mathfrak{sl}(2; \mathbb{C})\). On that complexified algebra we
have the basis introduced when we built the complexification,
\[
H = \begin{pmatrix} 1 & 0 \\\\ 0 & -1 \end{pmatrix}, \quad
E = \begin{pmatrix} 0 & 1 \\\\ 0 & 0 \end{pmatrix}, \quad
F = \begin{pmatrix} 0 & 0 \\\\ 1 & 0 \end{pmatrix},
\]
with brackets \([H, E] = 2E\), \([H, F] = -2F\), \([E, F] = H\). Substituting each into
the differential operator above gives
\[
\begin{align*}
\pi_m(H) &= -z_1 \frac{\partial}{\partial z_1} + z_2 \frac{\partial}{\partial z_2}, \\\\
\pi_m(E) &= -z_2 \frac{\partial}{\partial z_1}, \\\\
\pi_m(F) &= -z_1 \frac{\partial}{\partial z_2}.
\end{align*}
\]
These operators act transparently on the monomial basis. A direct computation on
\(z_1^{m-k} z_2^k\) yields
\[
\begin{align*}
\pi_m(H)\bigl(z_1^{m-k} z_2^k\bigr) &= (-m + 2k)\, z_1^{m-k} z_2^k, \\\\
\pi_m(E)\bigl(z_1^{m-k} z_2^k\bigr) &= -(m - k)\, z_1^{m-k-1} z_2^{k+1}, \\\\
\pi_m(F)\bigl(z_1^{m-k} z_2^k\bigr) &= -k\, z_1^{m-k+1} z_2^{k-1}.
\end{align*}
\]
The structure to notice is this. Each monomial \(z_1^{m-k} z_2^k\) is an
eigenvector of \(\pi_m(H)\), with eigenvalue \(-m + 2k\). As \(k\) runs from
\(0\) to \(m\) these eigenvalues run through \(-m, -m+2, \dots, m-2, m\), each
occurring once. The operator \(\pi_m(E)\) raises the power of \(z_2\) by one,
moving up the list of eigenvalues by \(2\), and annihilates the top monomial
\(z_2^m\) (the case \(k = m\)). The operator \(\pi_m(F)\) lowers the power of
\(z_2\) by one and annihilates the bottom monomial \(z_1^m\).
The diagonalizable generator \(H\), unavailable in the compact real form
\(\mathfrak{su}(2)\) and supplied precisely by complexification, thus grades
\(V_m\) into one-dimensional eigenspaces, with \(E\) and \(F\) stepping between
adjacent rungs. This grading is the lever on which the irreducibility argument
turns.
Irreducibility and the Weight Ladder
We can now prove that each \(\Pi_m\) is irreducible. The argument is purely a
matter of the eigenvalue grading just established. From any nonzero vector, the
operators \(E\) and \(F\) march along the rungs and reach every monomial, so no
proper subspace can be invariant. The eigenvalues of \(\pi_m(H)\) are called the
weights of the representation. The picture of the monomials
arranged by weight is its weight ladder, a single chain of rungs
joined by \(E\) and \(F\).
The following table records the ladder for \(V_m\). Each rung is a
one-dimensional weight space. The operator \(E\) climbs one rung, raising the
weight by \(2\), and \(F\) descends one, with each operator annihilating the rung
at the far end.
| Basis monomial |
Weight (eigenvalue of \(\pi_m(H)\)) |
Action of \(\pi_m(E)\) (climb) |
Action of \(\pi_m(F)\) (descend) |
| \(z_2^m\) (top) |
\(+m\) |
\(\mapsto 0\) (no higher rung) |
\(\mapsto\) nonzero \(\cdot\, z_1 z_2^{m-1}\) |
| \(\vdots\) |
\(\vdots\) |
climbs one rung |
descends one rung |
| \(z_1^{m-k} z_2^k\) |
\(-m + 2k\) |
\(\mapsto\) nonzero \(\cdot\, z_1^{m-k-1} z_2^{k+1}\) |
\(\mapsto\) nonzero \(\cdot\, z_1^{m-k+1} z_2^{k-1}\) |
| \(\vdots\) |
\(\vdots\) |
climbs one rung |
descends one rung |
| \(z_1^m\) (bottom) |
\(-m\) |
\(\mapsto\) nonzero \(\cdot\, z_1^{m-1} z_2\) |
\(\mapsto 0\) (no lower rung) |
Proposition (Irreducibility of \(\Pi_m\))
For each integer \(m \geq 0\), the representation \(\Pi_m\) of \(SU(2)\) on \(V_m\) is
irreducible.
Proof:
Since \(SU(2)\) is connected, \(\Pi_m\) is irreducible if and only if the associated
algebra representation \(\pi_m\) is irreducible, by the
group-algebra correspondence.
We prove the latter. It suffices to show that every nonzero invariant subspace
\(W \subseteq V_m\) is all of \(V_m\). So let \(W\) be a nonzero invariant subspace
and let \(w \in W\) be nonzero, written in the basis as
\[
w = a_0 z_1^m + a_1 z_1^{m-1} z_2 + \cdots + a_m z_2^m,
\]
with at least one coefficient nonzero. Let \(k_0\) be the smallest index with
\(a_{k_0} \neq 0\), so the lowest surviving term is \(a_{k_0} z_1^{m-k_0} z_2^{k_0}\).
Climbing to the top. Apply \(\pi_m(E)\) repeatedly. Each application
raises the power of \(z_2\) by one and annihilates any term already at the
top rung \(z_2^m\). Applied \(m - k_0\) times, it annihilates every term of
\(w\) except the \(k_0\) term, which is carried up to a nonzero multiple of
\(z_2^m\):
\[
\pi_m(E)^{\,m - k_0}\, w = c\, z_2^m, \quad c \neq 0.
\]
That the multiple is nonzero is exactly the statement that \(\pi_m(E)\) acting on
\(z_1^{m-k} z_2^k\) is a nonzero multiple of \(z_1^{m-k-1} z_2^{k+1}\) for every
\(k \lt m\), which the formulas above record. Since \(W\) is invariant and
\(w \in W\),
the vector \(c\, z_2^m\) lies in \(W\), and hence so does \(z_2^m\) itself.
Descending to recover the basis. Now apply \(\pi_m(F)\) repeatedly to
\(z_2^m\). Each application lowers the power of \(z_2\) by one and produces a nonzero
multiple of the next monomial down the ladder, so for every \(k\) with \(0 \leq k \leq
m\),
\[
\pi_m(F)^{\,k}\, z_2^m = c_k\, z_1^k z_2^{m-k}, \quad c_k \neq 0.
\]
As \(W\) is invariant and contains \(z_2^m\), it contains each \(z_1^k z_2^{m-k}\).
These are, up to reindexing, all \(m + 1\) basis monomials of \(V_m\). Therefore \(W\)
contains a basis of \(V_m\), so \(W = V_m\). No proper nonzero invariant subspace
exists, so \(\pi_m\) is irreducible, and hence so is \(\Pi_m\).
The proof used nothing beyond the eigenvalue grading: a single distinguished operator
whose eigenspaces are one-dimensional, together with two operators that move between
adjacent eigenspaces and terminate at the ends. This is the prototype of the
highest-weight method. The vector \(z_2^m\), of maximal weight \(+m\) and
annihilated by the raising operator \(E\), is a highest-weight vector.
From it the entire representation is generated by applying the lowering operator
\(F\), and the irreducibility is read off from the fact that this single chain of
lowerings reaches every basis vector.