07.07.05 · representation-theory / compact-lie

Representations of SU(2) and SO(3): the double cover, spin, and projective representations

shipped3 tiersLean: none

Anchor (Master): Sternberg *Group Theory and Physics* §5–§6; Weyl *The Theory of Groups and Quantum Mechanics* (1931) Ch. III–IV; Wigner *Group Theory and its Application to the Quantum Mechanics of Atomic Spectra* (1931/1959); Bargmann 1954 (Ann. Math. 59, 1–46) on the cohomological classification of projective representations

Intuition Beginner

Rotations in three-dimensional space form a group: you can do one rotation after another, and the result is again a rotation. This is the group called $S O (3)$ . It is the most physical symmetry there is — turn an object any way you like, and the laws of physics do not change. So you might expect that to describe how anything transforms under rotation, you only ever need the rotation group itself.

The surprise of the twentieth century is that nature needs a slightly bigger group. There is a group called $S U (2)$ that sits just above $S O (3)$ , wrapping around it twice. For every rotation, $S U (2)$ has two elements, not one. An electron remembers this doubling: rotate an electron by a full turn and its quantum state comes back with a minus sign, not back to where it started. You need two full turns to truly return. This is what physicists mean by spin one-half.

The everyday picture is the belt trick. Hold one end of a belt, give the other a single full twist, and the belt is tangled — you cannot smooth it out without turning the end again. Give it a second full twist and now you can untangle it by passing the belt over and around. One turn is not the identity; two turns is. That belt is feeling the difference between $S U (2)$ and $S O (3)$ .

Visual Beginner

A schematic of the two-to-one wrapping. On the right sits a small sphere of rotations, the group $S O (3)$ . On the left sits a larger sphere, the group $S U (2)$ , drawn as a covering that maps down onto the right sphere so that each point below is hit by exactly two points above — a north partner and a south partner that differ by a sign. A single loop drawn on the lower sphere (one full rotation) lifts to a path on the upper sphere that does not close up; only after going around twice does the lifted path return to its start.

The picture captures the whole story. Quantities that only feel the lower sphere are the integer-spin ones; they cannot tell the two partners apart. Quantities that feel the upper sphere — the spinors — change sign between a partner and its opposite, and that sign is exactly the half-integer spin.

Worked example Beginner

Take the simplest non-obvious case and watch the sign appear. A spin one-half state is described by a two-component column, and a rotation by angle $a$ about the vertical axis acts on it by the matrix

U (a) = (e^{- ia /2} 0 0 e^{+ ia /2}) .

Step 1. Put in a quarter turn, $a = 90$ degrees, which in radians is about $1.571$ . The top entry becomes $e^{- i \cdot 0.785}$ and the bottom becomes $e^{+ i \cdot 0.785}$ . The state has been rotated; nothing strange yet.

Step 2. Now put in a full turn, $a = 360$ degrees, which is $2 π$ radians. The top entry becomes $e^{- iπ} = - 1$ and the bottom becomes $e^{+ iπ} = - 1$ . So $U (36 0^{\circ}) = - I$ . A full rotation sends the state to minus itself.

Step 3. Put in two full turns, $a = 720$ degrees, which is $4 π$ . Now the top entry is $e^{- 2 π i} = + 1$ and the bottom is $e^{+ 2 π i} = + 1$ , so $U (72 0^{\circ}) = + I$ . Two full turns bring the state genuinely home.

What this tells us: the half-angle inside the matrix is the whole point. Because the angle is cut in half, a $360$ -degree rotation in real space becomes a $180$ -degree turn in the state, landing on minus one. The state lives upstairs on the bigger group, and it can tell apart a turn from a non-turn in a way that ordinary three-dimensional vectors cannot.

Check your understanding Beginner

Formal definition Intermediate+

Let $S U (2)$ denote the group of $2 \times 2$ complex matrices $g$ with $g^{†} g = I$ and $det g = 1$ . Every such matrix has the form $$ g = \begin{pmatrix} \alpha & -\bar\beta \ \beta & \bar\alpha \end{pmatrix}, \qquad |\alpha|^2 + |\beta|^2 = 1, $$ so $S U (2)$ is diffeomorphic to the unit three-sphere $S^{3}$ and is a compact, connected, simply connected Lie group of dimension three. Its Lie algebra $su (2)$ is the space of $2 \times 2$ skew-Hermitian traceless matrices, spanned over $R$ by $- \frac{i}{2} σ_{1}, - \frac{i}{2} σ_{2}, - \frac{i}{2} σ_{3}$ where $σ_{k}$ are the Pauli matrices.

Let $S O (3)$ denote the group of real $3 \times 3$ matrices $R$ with $R^{⊤} R = I$ and $det R = + 1$ , the rotation group of Euclidean three-space. It is compact and connected with fundamental group $π_{1} (S O (3)) = Z /2$ .

The covering homomorphism $φ : S U (2) \to S O (3)$ is built from the adjoint action on $su (2)$ . Identify $R^{3}$ with the traceless Hermitian matrices via $x = (x_{1}, x_{2}, x_{3}) \mapsto X = x_{1} σ_{1} + x_{2} σ_{2} + x_{3} σ_{3}$ , which satisfies $det X = - ∣ x ∣^{2}$ . For $g \in S U (2)$ the map $X \mapsto g X g^{- 1}$ preserves both tracelessness, Hermiticity, and the determinant, hence preserves $∣ x ∣^{2}$ ; it therefore defines an element $φ (g) \in S O (3)$ .

A finite-dimensional complex representation of $S U (2)$ is a continuous homomorphism $ρ : S U (2) \to G L (V)$ . For each non-negative half-integer $j \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ there is the spin- $j$ representation $D^{(j)}$ on the space $V_{j}$ of homogeneous polynomials of degree $2 j$ in two complex variables $z_{1}, z_{2}$ , with $g \in S U (2)$ acting by $(g \cdot p) (z) = p (g^{- 1} z)$ . The dimension of $V_{j}$ is $2 j + 1$ .

Counterexamples to common slips

$S O (3)$ is not simply connected. The loop of rotations about a fixed axis, swept from $0$ to $2 π$ , is a non-contractible loop generating $π_{1} (S O (3)) = Z /2$ . Its square is contractible, which is the homotopy-theoretic form of the belt trick.
Half-integer-spin $D^{(j)}$ are honest representations of $S U (2)$ , not of $S O (3)$ . The failure is not a defect of $D^{(j)}$ ; it is that $D^{(1/2)} (- I) = - I \neq = I$ , so $D^{(1/2)}$ is not constant on the two-element fibres of $φ$ and does not descend.
The kernel of $φ$ is exactly ${\pm I}$ , not all of the centre's powers. $S U (2)$ has centre ${\pm I} ≅ Z /2$ , and $φ$ collapses precisely that centre, giving $S O (3) ≅ S U (2) / {\pm I}$ .

Key theorem with proof Intermediate+

Theorem (the two-to-one cover and the descent criterion; Sternberg §5.2–§5.3). The map $φ : S U (2) \to S O (3)$ is a surjective Lie-group homomorphism with kernel ${\pm I}$ , hence a two-to-one covering and an isomorphism $S U (2) / {\pm I} \sim S O (3)$ . The irreducible representations of $S U (2)$ are exactly the $D^{(j)}$ for $j \in \frac{1}{2} Z_{\geq 0}$ , with $dim D^{(j)} = 2 j + 1$ , and $D^{(j)}$ descends to a representation of $S O (3)$ if and only if $j$ is an integer.

Proof. The construction of $φ$ already shows it is a homomorphism into the orthogonal group of the form $∣ x ∣^{2}$ , and continuity together with connectedness of $S U (2)$ forces the image into the identity component $S O (3)$ . For surjectivity, compute $φ$ on a one-parameter subgroup: with $g = exp (- \frac{i}{2} a σ_{3}) = diag (e^{- ia /2}, e^{ia /2})$ , conjugation $X \mapsto g X g^{- 1}$ rotates the $(x_{1}, x_{2})$ -plane by angle $a$ and fixes $x_{3}$ , so $φ (g)$ is the rotation by $a$ about the third axis. Every rotation is a rotation about some axis, and every axis is reached by an $S U (2)$ -conjugate of $σ_{3}$ , so $φ$ is onto.

For the kernel: $g \in ker φ$ means $g X g^{- 1} = X$ for all traceless Hermitian $X$ , hence $g$ commutes with $σ_{1}, σ_{2}, σ_{3}$ , which generate the full matrix algebra $M_{2} (C)$ . By Schur's lemma $g = λ I$ , and $det g = 1$ with $g \in S U (2)$ forces $λ = \pm 1$ . So $ker φ = {\pm I}$ , and the first isomorphism theorem gives $S O (3) ≅ S U (2) / {\pm I}$ . Because $S U (2) = S^{3}$ is simply connected while $π_{1} (S O (3)) = Z /2$ , the kernel of order two makes $φ$ a two-sheeted covering.

For the classification of irreducibles, pass to the complexified Lie algebra $su (2)_{C} ≅ sl_{2} (C)$ . The finite-dimensional irreducibles of $sl_{2} (C)$ are the highest-weight modules $V_{n}$ of highest weight $n \in Z_{\geq 0}$ and dimension $n + 1$ , built by a raising-lowering ladder. Setting $n = 2 j$ gives $dim = 2 j + 1$ . Since $S U (2)$ is compact, simply connected, the correspondence between Lie-algebra and Lie-group representations is a bijection, so each $V_{2 j}$ integrates to a unique $D^{(j)}$ on $S U (2)$ , and these exhaust the irreducibles by the highest-weight theorem.

For the descent criterion, a representation $ρ$ of $S U (2)$ factors through $φ$ to give a representation of $S O (3) = S U (2) / {\pm I}$ if and only if $ρ$ is constant on the fibres of $φ$ , equivalently $ρ (- I) = ρ (I) = id$ . On $V_{j}$ , the central element $- I$ acts on a degree- $2 j$ polynomial by $p (z) \mapsto p (- z) = (- 1)^{2 j} p (z)$ , so $D^{(j)} (- I) = (- 1)^{2 j} id$ . This equals the identity exactly when $2 j$ is even, that is when $j$ is an integer. Thus integer-spin $D^{(j)}$ descend to $S O (3)$ and half-integer-spin $D^{(j)}$ do not. $□$

Bridge. This theorem builds toward every physical use of angular momentum, and the descent criterion $D^{(j)} (- I) = (- 1)^{2 j}$ is the foundational reason the world splits into bosons and fermions. The central insight is that the obstruction to a half-integer spin living on $S O (3)$ is a single sign on a single central element, and that this sign is exactly the generator of $π_{1} (S O (3)) = Z /2$ detected by the belt trick. This pattern appears again in the tensor-product theory of the next section: putting these together, the Clebsch-Gordan rule for combining two spins is governed by the same $Z /2$ grading, since integer-plus-integer and half-plus-half give integer total spin while integer-plus-half gives half-integer. The bridge is the recognition that "representation of $S O (3)$ " and "representation of $S U (2)$ that is the identity on the centre" identify the integer-spin sector with the genuinely single-valued physics, and that this identifies the projective representations of $S O (3)$ with the honest representations of its double cover $S U (2)$ .

Exercises Intermediate+

Exercise 5 (medium, short-answer).

Use the character $χ_{j}$ and the descent criterion to explain why a spin- $\frac{1}{2}$ wavefunction, viewed as a section over $S O (3)$ , can only be defined up to a sign.

Hint

Compare $χ_{1/2} (a)$ at $a$ and at $a + 2 π$ ; the rotation $a + 2 π$ is the same element of $S O (3)$ as $a$ .

Answer

The rotations by $a$ and by $a + 2 π$ are equal as elements of $S O (3)$ but are the two distinct lifts in $S U (2)$ . On $D^{(1/2)}$ one computes $χ_{1/2} (a + 2 π) = - χ_{1/2} (a)$ , since the matrix at $a + 2 π$ equals minus the matrix at $a$ . So assigning a definite $S U (2)$ matrix to each $S O (3)$ rotation is impossible without a sign ambiguity, and the spinor is a section of a non-orientable association, defined up to the $\pm$ choice. This sign ambiguity is precisely a projective representation of $S O (3)$ with cocycle valued in ${\pm 1}$ . Rubric: full credit for the $a \to a + 2 π$ sign flip plus the projective-representation conclusion.

Exercise 6 (hard, short-answer).

Prove the Clebsch-Gordan series $D^{(j_{1})} \otimes D^{(j_{2})} = ⨁_{j = ∣ j_{1} - j_{2} ∣}^{j_{1} + j_{2}} D^{(j)}$ by a character computation.

Hint

Multiply $χ_{j_{1}} (a) χ_{j_{2}} (a)$ as Laurent polynomials in $u = e^{ia /2}$ , and show the product is the sum of the $χ_{j}$ for the stated range using the telescoping identity $χ_{j_{1}} χ_{j_{2}} = \sum χ_{j}$ .

Answer

Write $χ_{j} (a) = \sum_{m = - j}^{j} u^{2 m}$ with $u = e^{ia /2}$ . Then $$ \chi_{j_1}\chi_{j_2} = \sum_{m_1=-j_1}^{j_1}\sum_{m_2=-j_2}^{j_2} u^{2(m_1+m_2)}. $$ Grouping the lattice of pairs $(m_{1}, m_{2})$ by their total $m = m_{1} + m_{2}$ , the multiplicity of $u^{2 m}$ is the number of such pairs, which forms a symmetric "staircase" peaking at $m = 0$ . This staircase is exactly the sum of indicator profiles of the intervals $[- j, j]$ for $j = ∣ j_{1} - j_{2} ∣, ∣ j_{1} - j_{2} ∣ + 1, \dots, j_{1} + j_{2}$ , because each $χ_{j}$ contributes $+ 1$ to every $u^{2 m}$ with $∣ m ∣ \leq j$ . Matching multiplicities term by term gives $χ_{j_{1}} χ_{j_{2}} = \sum_{j = ∣ j_{1} - j_{2} ∣}^{j_{1} + j_{2}} χ_{j}$ . Since the irreducible characters of $S U (2)$ are linearly independent and a representation is determined by its character, this proves the direct-sum decomposition. Rubric: full credit for the character product, the staircase multiplicity argument, and the appeal to character-determines-representation.

Exercise 7 (hard, short-answer).

Show that the spin- $\frac{1}{2}$ projective representation of $S O (3)$ is not equivalent to any honest linear representation, but that it lifts to an honest representation of $S U (2)$ .

Hint

Suppose a continuous map $R \mapsto \tilde{ρ} (R) \in U (2)$ with $\tilde{ρ} (R) \tilde{ρ} (R^{'}) = \tilde{ρ} (R R^{'})$ existed, agreeing with the spin matrices up to phase. Use simple connectivity of $S U (2)$ and the order of $- I$ .

Answer

A projective representation of $S O (3)$ is a continuous $\overset{ρ}{ˉ} : S O (3) \to P U (2)$ together with a chosen lift to $U (2)$ that satisfies $ρ (R) ρ (R^{'}) = c (R, R^{'}) ρ (R R^{'})$ for a phase cocycle $c$ . The spin- $\frac{1}{2}$ matrices give such a lift with $c$ valued in ${\pm 1}$ . If an honest linear representation existed, the cocycle would be a coboundary, so $c$ could be removed by rephasing; but rephasing changes each $ρ (R)$ only by a scalar and cannot turn the order-four element representing a $2 π$ rotation (which squares to $+ I$ only after two turns) into an order-two assignment over $S O (3)$ . Concretely, the loop of $2 π$ rotations is non-contractible in $S O (3)$ , so any continuous single-valued lift would assign it a sign that must equal both $+ 1$ (by closing the loop in $S O (3)$ ) and $- 1$ (by the spin matrices), a contradiction. Pulling back along $φ$ to the simply connected $S U (2)$ removes the obstruction, since $π_{1} (S U (2)) = 0$ kills the cocycle and the spin- $\frac{1}{2}$ representation becomes the honest $D^{(1/2)}$ . Rubric: full credit for the cocycle/coboundary argument plus the simple-connectivity lift.

Advanced results Master

Theorem (the spin- $j$ irreducibles as symmetric powers). For each $j \in \frac{1}{2} Z_{\geq 0}$ , the representation $D^{(j)}$ is the $2 j$ -th symmetric power $Sym^{2 j} (C^{2})$ of the defining representation $D^{(1/2)} = C^{2}$ . Equivalently $V_{j}$ is the space of degree- $2 j$ homogeneous polynomials in two variables, on which the symmetric-power structure is the polynomial multiplication. These are pairwise inequivalent and exhaust $S U (2)$ .

The defining representation $C^{2}$ is self-dual via the symplectic form $ε (z, w) = z_{1} w_{2} - z_{2} w_{1}$ , so every $D^{(j)}$ is self-dual: $D^{(j)} ≅ (D^{(j)})^{*}$ . The Frobenius-Schur indicator equals $+ 1$ for integer $j$ (real, orthogonal type) and $- 1$ for half-integer $j$ (quaternionic, symplectic type). This real-versus-quaternionic dichotomy is the same parity that governs descent to $S O (3)$ , and it is the representation-theoretic origin of the Kramers degeneracy of half-integer-spin systems under time reversal.

Theorem (Clebsch-Gordan with explicit coefficients). The decomposition $D^{(j_{1})} \otimes D^{(j_{2})} = ⨁_{j = ∣ j_{1} - j_{2} ∣}^{j_{1} + j_{2}} D^{(j)}$ holds with multiplicity one in each summand. The change of basis from the uncoupled basis $∣ j_{1} m_{1} ⟩ \otimes ∣ j_{2} m_{2} ⟩$ to the coupled basis $∣ j m ⟩$ is implemented by the Clebsch-Gordan coefficients $⟨ j_{1} m_{1} j_{2} m_{2} ∣ j m ⟩$ , which are real, vanish unless $m = m_{1} + m_{2}$ and $∣ j_{1} - j_{2} ∣ \leq j \leq j_{1} + j_{2}$ , and satisfy the Racah orthogonality relations.

The multiplicity-one feature is special to $S U (2)$ among the simple compact groups and is what makes the angular-momentum addition rules so rigid: a tensor product of two irreducibles never repeats a summand. For three or more factors the recoupling is encoded by the Wigner $6 j$ and $9 j$ symbols, and the associativity of the tensor product becomes the Biedenharn-Elliott pentagon identity, the same pentagon that governs fusion categories.

Theorem (Wigner's theorem and the origin of projective representations; Wigner 1931, Bargmann 1954). Every symmetry of a quantum system — a bijection of rays preserving transition probabilities — is implemented on Hilbert space by an operator that is either unitary or antiunitary, unique up to a phase. Consequently a symmetry group $G$ acts not by a linear representation but by a projective representation $G \to P U (H)$ . For $G = S O (3)$ the obstruction to lifting to an honest linear representation is classified by $H^{2} (S O (3); U (1))$ , and because $π_{1} (S O (3)) = Z /2$ this obstruction is exactly two-valued: every projective representation of $S O (3)$ lifts to an honest representation of its universal cover $S U (2)$ .

The general principle, due to Bargmann, is that for a connected Lie group $G$ the continuous projective unitary representations are in bijection with the honest unitary representations of a central extension determined by $H^{2} (g; R)$ together with $π_{1} (G)$ . For semisimple $G$ the Lie-algebra cohomology vanishes, so the only obstruction is topological — the fundamental group — and the universal cover absorbs all projective representations. $S O (3)$ is the cleanest nonabelian example: its single topological obstruction $Z /2$ is the entire reason half-integer spin exists.

Theorem (Peter-Weyl realization for $S U (2)$ ). The matrix coefficients of the $D^{(j)}$ form a complete orthogonal system in $L^{2} (S U (2))$ , and as a representation of $S U (2) \times S U (2)$ , $$ L^2(SU(2)) ;\cong; \widehat{\bigoplus_{j \in \frac12 \mathbb{Z}{\geq 0}}} ; D^{(j)} \otimes (D^{(j)})^*. $$ *Restricting to functions invariant under the centre recovers the Peter-Weyl decomposition of $L^{2} (S O (3))$ , which contains only the integer-spin blocks; these are realized concretely as the spherical harmonics $Y\ell^m $s p annin g t h e$ D^{(\ell)} $- i so t y p i c p a r t o f$ L^2(S^2)$.*

Synthesis. The double cover $S U (2) \to S O (3)$ is the foundational reason quantum mechanics needs more than the rotation group, and the central insight is that a single sign — the value $(- 1)^{2 j}$ of $D^{(j)}$ on the central element $- I$ — simultaneously controls descent to $S O (3)$ , the boson-fermion dichotomy, the real-versus-quaternionic type of the representation, and the obstruction class in $H^{2} (S O (3); U (1))$ . This is exactly the topological content of $π_{1} (S O (3)) = Z /2$ made into representation theory: putting these together, the half-integer-spin representations are precisely the honest representations of the universal cover that fail to descend, and the projective representations of $S O (3)$ are identified with the linear representations of $S U (2)$ . The Clebsch-Gordan series generalises this parity bookkeeping to products of spins, while Wigner's theorem supplies the physical reason projective representations are unavoidable, and Bargmann's cohomological classification shows that for a semisimple group the universal cover absorbs every projective phase. The bridge between the topology and the spectroscopy is the character formula, which appears again in the Weyl character formula for every compact group, with $S U (2)$ as its base case. The whole arc — from the belt trick to the periodic table's spin-orbit fine structure — is one statement: representations of the symmetry that fixes the laws of physics are governed not by $S O (3)$ but by its simply connected double cover.

Full proof set Master

Proposition (descent criterion, complete). A finite-dimensional representation $ρ$ of $S U (2)$ descends through $φ$ to a representation of $S O (3)$ if and only if $ρ (- I) = id$ . In particular $D^{(j)}$ descends if and only if $j \in Z$ .

Proof. A representation $ρ : S U (2) \to G L (V)$ factors as $ρ = \overset{ρ}{ˉ} \circ φ$ for some $\overset{ρ}{ˉ} : S O (3) \to G L (V)$ if and only if $ρ$ is constant on each fibre $φ^{- 1} (R)$ . The fibres are the cosets of $ker φ = {\pm I}$ , so constancy on fibres is equivalent to $ρ (g) = ρ (- g)$ for all $g$ , equivalently $ρ (- I) ρ (g) = ρ (g)$ for all $g$ , equivalently $ρ (- I) = id$ . When such $\overset{ρ}{ˉ}$ exists it is unique and continuous because $φ$ is a surjective open map (a covering), so $\overset{ρ}{ˉ} = ρ \circ s$ for any local section $s$ is well-defined independent of the section. For $D^{(j)}$ on the degree- $2 j$ polynomials, $- I$ acts by $p (z) \mapsto p ((- I)^{- 1} z) = p (- z) = (- 1)^{2 j} p (z)$ , so $D^{(j)} (- I) = (- 1)^{2 j} id$ , which is the identity precisely when $2 j$ is even. $□$

Proposition (irreducibility of $D^{(j)}$ ). Each $D^{(j)}$ is an irreducible representation of $S U (2)$ .

Proof. It suffices to prove irreducibility of $V_{2 j}$ as a module over $sl_{2} (C) = ⟨ e, f, h ⟩$ , since an $S U (2)$ -invariant subspace is $su (2)$ -invariant and hence $sl_{2} (C)$ -invariant by complexification, and conversely an invariant complex subspace integrates back. In the polynomial model take $h = z_{1} \partial_{z_{1}} - z_{2} \partial_{z_{2}}$ , $e = z_{1} \partial_{z_{2}}$ , $f = z_{2} \partial_{z_{1}}$ . The monomials $v_{k} = z_{1}^{2 j - k} z_{2}^{k}$ for $k = 0, \dots, 2 j$ are $h$ -eigenvectors with distinct eigenvalues $2 j - 2 k$ . Any nonzero invariant subspace $W$ is $h$ -stable, hence spanned by a subset of the $v_{k}$ (distinct eigenvalues force eigenvector containment). If $v_{k} \in W$ then $e v_{k}$ and $f v_{k}$ are nonzero multiples of $v_{k - 1}$ and $v_{k + 1}$ within range, so repeated application of $e$ and $f$ reaches every $v_{ℓ}$ . Thus $W = V_{2 j}$ , and $D^{(j)}$ is irreducible. $□$

Proposition (completeness of the $D^{(j)}$ ). Every finite-dimensional irreducible complex representation of $S U (2)$ is equivalent to some $D^{(j)}$ .

Proof. Let $ρ$ be a finite-dimensional irreducible representation of $S U (2)$ on $V$ . Differentiating gives a representation of $su (2)$ , which complexifies to a representation of $sl_{2} (C)$ on $V$ . By the highest-weight theory of $sl_{2} (C)$ , the operator $h$ has a highest eigenvector $v$ annihilated by $e$ , with eigenvalue a non-negative integer $n$ ; the vectors $v, f v, f^{2} v, \dots, f^{n} v$ span an $(n + 1)$ -dimensional irreducible submodule, which by irreducibility of $V$ is all of $V$ . This submodule is isomorphic to $V_{n}$ , and since $S U (2)$ is connected and simply connected the Lie-algebra isomorphism integrates to an isomorphism of group representations $V ≅ D^{(n /2)}$ . Setting $j = n /2$ exhibits $ρ ≅ D^{(j)}$ . $□$

Proposition (self-duality and Frobenius-Schur type). Each $D^{(j)}$ is self-dual, with Frobenius-Schur indicator $ν (D^{(j)}) = (- 1)^{2 j}$ , so integer spins are of real (orthogonal) type and half-integer spins are of quaternionic (symplectic) type.

Proof. The symplectic form $ε$ on $C^{2}$ is $S U (2)$ -invariant because $S U (2) = S p (1)$ , giving an isomorphism $C^{2} ≅ (C^{2})^{*}$ and hence $D^{(j)} = Sym^{2 j} (C^{2}) ≅ Sym^{2 j} ((C^{2})^{*}) ≅ (D^{(j)})^{*}$ . The Frobenius-Schur indicator is $ν = \int_{S U (2)} χ_{j} (g^{2}) d g$ . Using $χ_{j} (g^{2})$ expanded in the character basis and the orthogonality relations, the integral evaluates to $+ 1$ when the invariant bilinear form on $D^{(j)}$ is symmetric and $- 1$ when it is alternating. The invariant form on $Sym^{2 j} (C^{2})$ induced by $ε$ is symmetric for even $2 j$ and alternating for odd $2 j$ , since $ε$ is alternating and the symmetric power of an alternating form has parity $(- 1)^{2 j}$ . Hence $ν (D^{(j)}) = (- 1)^{2 j}$ . $□$

Connections Master

Representations of $sl_{2} (C)$ 07.06.11. The classification of the $D^{(j)}$ rests entirely on the highest-weight ladder for $sl_{2} (C) = su (2)_{C}$ : the integer highest weight $n = 2 j$ and the $(n + 1)$ -dimensional irreducible are the algebraic skeleton onto which the present unit hangs the group-theoretic and topological story. The descent criterion and the half-integer obstruction are invisible at the Lie-algebra level — they are exactly the global information the algebra cannot see, which is why the double cover must be analysed at the group level.
Angular momentum operators and SU(2) representations 12.05.01. The physics-facing construction of the $D^{(j)}$ from raising and lowering operators $J_{\pm}$ and the eigenvalue $j (j + 1)$ of the Casimir is the same representation theory computed in the ladder basis. This unit supplies the structural reason those operators close into $S U (2)$ rather than $S O (3)$ , and why the spin quantum number can be half-integral; the angular-momentum unit supplies the explicit matrix elements and the spectroscopic vocabulary.
Spin group 03.09.03. $S U (2) = Spin (3)$ is the lowest-dimensional spin group, and the covering $φ : S U (2) \to S O (3)$ is the rank-three case of the general $Spin (n) \to S O (n)$ double cover built from the Clifford algebra. The projective-representation phenomenon isolated here is the representation-theoretic shadow of the spinor representations of $Spin (n)$ , and the present unit is the concrete model that the abstract spin-geometry machinery specialises to in three dimensions.
Stern-Gerlach and spin-1/2 12.01.02. The empirical two-valuedness of the electron — the split beam, the sign after a full rotation — is the physical manifestation of $D^{(1/2)} (- I) = - I$ . This unit proves that the two-valuedness is forced by the topology of $S O (3)$ rather than being an ad-hoc postulate, closing the loop between the experiment and the covering-group mathematics.
Compact Lie group representation 07.07.01. The general theory of compact-group representations — complete reducibility, the existence of an invariant inner product, the orthogonality of matrix coefficients — is what guarantees the $D^{(j)}$ are unitarizable and that $L^{2} (S U (2))$ decomposes into their isotypic blocks. $S U (2)$ is the smallest nonabelian instance where the abstract Peter-Weyl statement becomes a fully explicit spherical-harmonic expansion.

Historical & philosophical context Master

The double-valued representations of the rotation group entered physics through the spinning electron. Wolfgang Pauli introduced the two-component spin wavefunction in 1927 (Z. Phys. 43, 601) to encode the electron's two-valuedness, writing down what are now the Pauli matrices without yet identifying the underlying group structure. Hermann Weyl, in Gruppentheorie und Quantenmechanik (Hirzel, Leipzig 1928; English ed. 1931) ^{[Weyl 1931]}, gave the systematic representation-theoretic treatment, recognizing the spin representations as the double-valued representations of the proper rotation group and connecting them to the unimodular unitary group in two variables.

Eugene Wigner's 1931 monograph Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg; English ed. Academic Press 1959) ^{[Wigner 1931]} built the $D^{(j)}$ explicitly as the irreducible representations of the rotation group and made angular-momentum coupling — the Clebsch-Gordan series and the coefficients now bearing the name of the nineteenth-century invariant theorists Alfred Clebsch and Paul Gordan — into the working language of atomic spectroscopy. Wigner's later theorem on the unitary-or-antiunitary implementation of symmetries, sharpened by Valentine Bargmann in On unitary ray representations of continuous groups (Ann. of Math. 59 (1954) 1–46) ^{[Bargmann 1954]}, identified the precise sense in which quantum symmetry groups act projectively, and reduced the lifting question for connected Lie groups to a computation in Lie-algebra cohomology together with the fundamental group. For $S O (3)$ the cohomology vanishes and only $π_{1} = Z /2$ survives, which is why half-integer spin exists at all. Élie Cartan had constructed spinors abstractly in 1913 (Bull. Soc. Math. France 41, 53) ^{[Cartan 1913]} from the geometry of isotropic subspaces, two decades before the physical motivation arrived; the convergence of Cartan's geometry, Weyl's representation theory, and Pauli's electron is one of the recorded instances where a mathematical structure preceded the physics it would describe. Sternberg's Group Theory and Physics (Cambridge 1994) presents this arc in a single uniform notation, from the covering homomorphism to the Wigner classification.

Bibliography Master

@book{Sternberg1994GroupTheoryPhysics,
  author    = {Sternberg, Shlomo},
  title     = {Group Theory and Physics},
  publisher = {Cambridge University Press},
  year      = {1994}
}

@book{Weyl1931GroupsQM,
  author    = {Weyl, Hermann},
  title     = {The Theory of Groups and Quantum Mechanics},
  publisher = {Methuen (English translation by H. P. Robertson)},
  year      = {1931},
  note      = {German original: Gruppentheorie und Quantenmechanik, Hirzel, Leipzig 1928}
}

@book{Wigner1959AtomicSpectra,
  author    = {Wigner, Eugene P.},
  title     = {Group Theory and its Application to the Quantum Mechanics of Atomic Spectra},
  publisher = {Academic Press},
  year      = {1959},
  note      = {German original: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Vieweg, Braunschweig 1931}
}

@article{Bargmann1954RayReps,
  author  = {Bargmann, Valentine},
  title   = {On unitary ray representations of continuous groups},
  journal = {Ann. of Math. (2)},
  volume  = {59},
  year    = {1954},
  pages   = {1--46}
}

@article{Pauli1927Spin,
  author  = {Pauli, Wolfgang},
  title   = {Zur Quantenmechanik des magnetischen Elektrons},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {43},
  year    = {1927},
  pages   = {601--623}
}

@article{Cartan1913Spineurs,
  author  = {Cartan, {\'E}lie},
  title   = {Les groupes projectifs qui ne laissent invariante aucune multiplicit{\'e} plane},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {41},
  year    = {1913},
  pages   = {53--96}
}

@book{Hall2013QTM,
  author    = {Hall, Brian C.},
  title     = {Quantum Theory for Mathematicians},
  publisher = {Springer},
  series    = {Graduate Texts in Mathematics},
  volume    = {267},
  year      = {2013}
}

@book{BrockerTomDieck1985,
  author    = {Br{\"o}cker, Theodor and tom Dieck, Tammo},
  title     = {Representations of Compact Lie Groups},
  publisher = {Springer},
  series    = {Graduate Texts in Mathematics},
  volume    = {98},
  year      = {1985}
}

Prerequisites

07.07.01
07.06.11
12.05.01
03.09.03
12.01.02

Tier anchors

beginner: Sternberg *Group Theory and Physics* §5 informal opening on rotations and the belt trick; 3Blue1Brown 'Quaternions and 3d rotation' for the double-cover picture
intermediate: Sternberg *Group Theory and Physics* §5.2–§5.5 (the $D^{(j)}$, the covering homomorphism, Clebsch-Gordan); Hall *Quantum Theory for Mathematicians* §16–§17
master: Sternberg *Group Theory and Physics* §5–§6; Weyl *The Theory of Groups and Quantum Mechanics* (1931) Ch. III–IV; Wigner *Group Theory and its Application to the Quantum Mechanics of Atomic Spectra* (1931/1959); Bargmann 1954 (Ann. Math. 59, 1–46) on the cohomological classification of projective representations

References

Sternberg — Group Theory and Physics · §5.1–§5.5 (SU(2) irreducibles D^{(j)}, the covering SU(2)→SO(3), half-integer spin), §5.6 (Clebsch-Gordan series and coefficients), §6 (Wigner's theorem and projective representations)
Hall — Quantum Theory for Mathematicians · §16 (the Lie algebra su(2) and its representations), §17 (the universal cover, projective representations, and the appearance of SU(2) in quantum mechanics)
Weyl — The Theory of Groups and Quantum Mechanics · Ch. III–IV; the rotation group, its representations, and the double-valued (spinor) representations of the proper orthogonal group in three variables
Wigner — Group Theory and its Application to the Quantum Mechanics of Atomic Spectra · Ch. 15 (the representations of the three-dimensional rotation group D^{(j)}), Ch. 20 (the unitary–antiunitary theorem and the role of the covering group)
Bargmann — On unitary ray representations of continuous groups · Ann. of Math. 59 (1954) 1–46; the cohomological obstruction theory classifying when a projective representation lifts to a true representation of the universal cover

Estimated time

beginner: 18m
intermediate: 40m
master: 80m