Examples of Representations

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Introduction Standard, Trivial, and Adjoint The Representations of \(SU(2)\) Irreducibility and the Weight Ladder

Introduction

We have the vocabulary of representation theory in hand: a representation of a matrix Lie group is a continuous homomorphism into the invertible linear maps of a finite-dimensional 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 with a nonzero Lie algebra 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 through it that of \(SO(3)\). The distinguished operator is the generator \(H\), diagonal with real eigenvalues, and the operators that step between its eigenspaces 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

Every matrix Lie group with a nonzero Lie algebra carries three representations 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 \(\mathfrak{g}\), a real vector space. Since \(AXA^{-1}\) depends continuously on \(A\), this is a real representation of \(G\), and a complex one whenever \(\mathfrak{g}\) happens to be a complex subspace of \(M_n(\mathbb{C})\). 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. The two are in fact isomorphic as representations. The isomorphism is the hat map \(\boldsymbol{\omega} \mapsto \hat{\boldsymbol{\omega}}_{\times}\) that identifies \(\mathbb{R}^3\) with \(\mathfrak{so}(3)\) and carries the cross product to the bracket. That it intertwines the two actions is the identity \(R\,\hat{\boldsymbol{\omega}}_{\times}\,R^{-1} = \widehat{R\boldsymbol{\omega}}_{\times}\) for \(R \in SO(3)\), which rests on rotations preserving the cross product, a fact we take for granted here. This agreement is special to \(n = 3\) among the groups \(SO(n)\) with \(n \geq 2\). The adjoint representation of \(SO(n)\) has the dimension \(n(n-1)/2\) of the group rather than the dimension \(n\) of the space it was presented on, and the two dimensions are equal only in that case.

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, which for \(m \geq 1\) is not a representation. Two further points close the verification. Since \(\Pi_m(I) = I\), each \(\Pi_m(U)\) is invertible with inverse \(\Pi_m(U^{-1})\), so \(\Pi_m\) takes values in \(GL(V_m)\). And the entries of \(\Pi_m(U)\) in the monomial basis are polynomials in the entries of \(U^{-1} = U^*\), so \(\Pi_m\) is continuous. Hence \(\Pi_m\) is 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 \(\mathfrak{sl}(2, \mathbb{C})\). The hypothesis of that extension, \(\mathfrak{su}(2) \cap i\,\mathfrak{su}(2) = \{0\}\), holds because a matrix that is both skew-Hermitian and Hermitian is zero, and since the differential operator above is complex-linear in the entries of \(W\), the extension is given 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 generator \(H\), whose eigenvalues are real, is not in the compact real form \(\mathfrak{su}(2)\), whose elements are skew-Hermitian and so have purely imaginary eigenvalues. The grading by its eigenspaces is already present there, since \(-iH\) lies in \(\mathfrak{su}(2)\) and \(\pi_m(-iH) = -i\,\pi_m(H)\) has the same one-dimensional eigenspaces. What complexification adds is the pair of operators \(E\) and \(F\), which are not in \(\mathfrak{su}(2)\) and which step between adjacent rungs. The grading and these two operators together are 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 nonzero 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 \(V_m\) with \(m \geq 1\). 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. For \(m = 0\) the ladder is the single rung spanned by the constant polynomial \(1\), which both \(E\) and \(F\) annihilate.

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:

The group \(SU(2)\) is connected. Indeed, for \(U \in SU(2)\) the conditions \(U^{-1} = U^*\) and \(\det U = 1\) force \(U = \begin{pmatrix} a & -\overline{b} \\ b & \overline{a} \end{pmatrix}\) with \(|a|^2 + |b|^2 = 1\), which identifies \(SU(2)\) with the unit sphere in \(\mathbb{C}^2\), a path-connected set. Hence \(\Pi_m\) 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 \(F\). The irreducibility is read off from two facts. Repeated raising shows that every nonzero invariant subspace contains \(z_2^m\), and this single chain of lowerings from \(z_2^m\) reaches every basis vector.