Representation Theory

1. Representations of Groups

Every group has at least two representations:

• The trivial representation: $\rho(g) = 1$ for all $g$ (one-dimensional, every element maps to the identity)
• The regular representation: $G$ acts on the vector space of functions on $G$ by left multiplication

For Lie groups, we also have representations of the Lie algebra:

The connection to the group representation is via the exponential map:$D(e^X) = e^{\rho(X)}$.

The adjoint representation is especially important. For a Lie algebra $\mathfrak{g}$, the adjoint representation acts on $\mathfrak{g}$ itself:

The adjoint representation has dimension equal to the dimension of the algebra. For$\mathrm{SU}(3)$, the adjoint is the $\mathbf{8}$ representation — this is precisely the representation under which gluons transform.

2. Irreducible Representations

A subspace $W \subseteq V$ is invariant under a representation $\rho$ if$\rho(g)w \in W$ for all $g \in G$, $w \in W$. A representation is irreducible (or an irrep) if the only invariant subspaces are$\{0\}$ and $V$ itself.

Irreducible representations are the building blocks: any representation can be decomposed into irreps. For finite groups and compact Lie groups, every representation is equivalent to a unitary one (Weyl's unitarity trick), and every unitary representation decomposes as a direct sum of irreps:

where $V^{(j)}$ are irreducible and $n_j$ are multiplicities. This is complete reducibility (Maschke's theorem for finite groups, Peter-Weyl theorem for compact Lie groups).

Example: SU(2) irreps. The irreducible representations of $\mathrm{SU}(2)$ are labeled by $j = 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$ with dimension $2j + 1$. The representation space has a basis $|j, m\rangle$ with $m = -j, \ldots, j$. The generators act as:

The Casimir operator $\mathbf{J}^2 = J_1^2 + J_2^2 + J_3^2$ takes the value $j(j+1)$on the entire irrep. By Schur's lemma (below), this is the only possibility for an operator that commutes with all generators.

3. Schur's Lemma

Schur's lemma is the most important single result in representation theory. It has two parts:

Proof. Consider $\ker T \subseteq V_1$. For any $v \in \ker T$:$T(\rho_1(g)v) = \rho_2(g)(Tv) = 0$, so $\rho_1(g)v \in \ker T$. Thus$\ker T$ is an invariant subspace of $V_1$. Since $\rho_1$ is irreducible,$\ker T = \{0\}$ or $\ker T = V_1$. Similarly, $\mathrm{im}(T)$ is an invariant subspace of $V_2$, so it is either $\{0\}$ or $V_2$. If$\ker T = \{0\}$ and $\mathrm{im}(T) = V_2$, then $T$ is an isomorphism. Otherwise $T = 0$. $\square$

Proof. Since we are over $\mathbb{C}$, $T$ has at least one eigenvalue$\lambda$. Then $T - \lambda I$ also commutes with all $\rho(g)$ and has a nontrivial kernel. By Part I (with $V_1 = V_2 = V$), it must be zero. Hence$T = \lambda I$. $\square$

Physical consequence: In quantum mechanics, if $H$ commutes with all generators of a symmetry group, and the Hilbert space carries an irreducible representation, then $H = E \cdot I$ on that space — all states in a multiplet are degenerate. This is why the $2j+1$ magnetic substates $|j,m\rangle$ all have the same energy when rotational symmetry is unbroken.

4. Character Theory

The character of a representation $\rho$ is the function:

Characters are powerful because they are class functions (constant on conjugacy classes: $\chi(hgh^{-1}) = \chi(g)$) and uniquely determine the representation up to equivalence. The key orthogonality relations are:

For finite groups of order $|G|$:

For compact Lie groups, the sum becomes an integral over the group (with Haar measure):

For $\mathrm{SU}(2)$, an element in the spin-$j$ representation rotated by angle$\theta$ about the $z$-axis has character:

This is the Weyl character formula specialized to $\mathrm{SU}(2)$. It allows us to decompose tensor products by simply multiplying characters and expanding in the basis of irreducible characters.

To find how many times an irrep $j$ appears in a representation with character $\chi$:

Character tables for finite groups are square matrices whose rows are irreducible characters and whose columns are conjugacy classes. The orthogonality relations make the character table a unitary matrix (up to normalization). The number of irreps equals the number of conjugacy classes, and the sum of squares of dimensions equals the group order:

5. Tensor Products and Clebsch-Gordan Decomposition

Given two representations $V_1$ and $V_2$, the tensor product representationacts on $V_1 \otimes V_2$ as:

The tensor product is generally reducible. The decomposition into irreps is the Clebsch-Gordan decomposition.

For $\mathrm{SU}(2)$, coupling spin $j_1$ and spin $j_2$:

The change-of-basis coefficients are the Clebsch-Gordan coefficients:

with the constraint $m = m_1 + m_2$. These coefficients are computed recursively using the raising/lowering operators.

Derivation of CG coefficients. Start with the highest weight state$|j_1+j_2, j_1+j_2\rangle = |j_1, j_1\rangle \otimes |j_2, j_2\rangle$. Apply $J_- = J_-^{(1)} + J_-^{(2)}$repeatedly to generate all states in the $j = j_1 + j_2$ multiplet. The state orthogonal to $|j_1+j_2, j_1+j_2-1\rangle$ in the $m = j_1+j_2-1$ subspace starts the$j = j_1 + j_2 - 1$ multiplet. Continue until all states are accounted for.

Dimension check: The tensor product space has dimension $(2j_1+1)(2j_2+1)$. The direct sum gives $\sum_{j=|j_1-j_2|}^{j_1+j_2} (2j+1) = (2j_1+1)(2j_2+1)$. This identity can be verified directly.

Example: $\frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 1$. Two spin-1/2 particles combine into a singlet (total spin 0) and a triplet (total spin 1). In terms of states:

6. Young Tableaux for SU(N)

For $\mathrm{SU}(N)$ with $N \geq 3$, the representation theory is richer than$\mathrm{SU}(2)$. Young diagrams provide a systematic way to classify all irreducible representations.

A Young diagram is a collection of boxes arranged in left-justified rows, with row lengths non-increasing: $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k > 0$. The partition $\lambda = (\lambda_1, \ldots, \lambda_k)$ with at most $N-1$ rows labels an irrep of $\mathrm{SU}(N)$.

The dimension of the representation is given by the hook length formula:

where $h(i,j)$ is the hook length at box $(i,j)$ — the number of boxes directly to the right plus directly below plus one (for the box itself).

Key examples for SU(3):

• $\lambda = (1)$ (one box): the fundamental representation $\mathbf{3}$ (quarks)
• $\lambda = (1,1)$ (one column of 2): the conjugate $\bar{\mathbf{3}}$ (antiquarks)
• $\lambda = (2)$ (two boxes in a row): dimension 6, the symmetric $\mathbf{6}$
• $\lambda = (2,1)$ (L-shape): the adjoint $\mathbf{8}$ (gluons, mesons)
• $\lambda = (3)$ (three boxes in a row): dimension 10, the decuplet $\mathbf{10}$ (baryons like $\Delta, \Sigma^*, \Xi^*, \Omega^-$)

Tensor product decomposition using Young tableaux proceeds by the Littlewood-Richardson rule: to compute $\lambda \otimes \mu$, fill the boxes of $\mu$with labels and append them to $\lambda$ following specific rules. For example:

The first decomposition shows that two quarks can form a symmetric sextet or an antisymmetric antitriplet. The second shows that a quark-antiquark pair forms an octet (mesons) plus a singlet.

For baryons (three quarks), the color decomposition is:

The singlet $\mathbf{1}$ (totally antisymmetric in color) is the physical baryon state, enforcing color confinement.

7. Weights and Root Systems

The classification of representations uses the concepts of weights and roots. Choose a maximal set of commuting generators $H_i$ (the Cartan subalgebra). A weight is the set of simultaneous eigenvalues:

where $r$ is the rank of the algebra. Roots are the non-zero weights of the adjoint representation. For each root $\alpha$, there is a raising/lowering operator $E_\alpha$ satisfying:

The highest weight of an irreducible representation is the weight from which no further raising is possible. The highest weight uniquely determines the irrep and can be expanded in the basis of fundamental weights: $\Lambda = \sum_i a_i \omega_i$ where$a_i \in \mathbb{Z}_{\geq 0}$ are the Dynkin labels.

For $\mathrm{SU}(2)$ (rank 1): there is one Cartan generator $J_3$, roots$\pm 1$, and the spin-$j$ representation has highest weight $j$ with Dynkin label $[2j]$. For $\mathrm{SU}(3)$ (rank 2): the Dynkin labels $(p,q)$correspond to the standard labeling of SU(3) irreps.

The Weyl dimension formula gives the dimension of any irrep from its highest weight:

where $\rho$ is the Weyl vector (half the sum of positive roots) and the product runs over all positive roots.

8. Branching Rules and Subgroup Decomposition

When a symmetry group $G$ is broken to a subgroup $H \subset G$, irreducible representations of $G$ decompose into (generally reducible) representations of $H$. These branching rules are central to understanding symmetry breaking.

Example: SU(3) $\to$ SU(2) $\times$ U(1). The fundamental representation decomposes as:

The subscripts denote the U(1) hypercharge. The adjoint decomposes as:

This shows how the meson octet splits into isospin multiplets: the pion triplet, two kaon doublets, and the eta singlet. Branching rules are essential in grand unified theories where $\mathrm{SU}(5)$ or $\mathrm{SO}(10)$ breaks to the Standard Model group.

Selection rules from representation theory. A transition matrix element$\langle f | \hat{O} | i \rangle$ is nonzero only if the final state representation appears in the tensor product of the operator representation with the initial state:

For electric dipole transitions (operator transforms as a vector, $j=1$), this gives the selection rule $\Delta j = 0, \pm 1$ (excluding $0 \to 0$).

9. The Wigner-Eckart Theorem

The Wigner-Eckart theorem connects representation theory directly to physical observables. A tensor operator $T_q^{(k)}$ of rank $k$ transforms under rotations as:

The theorem states:

The matrix element factorizes into a Clebsch-Gordan coefficient (purely geometric, encoding the angular dependence) and a reduced matrix element $\langle j' \| T^{(k)} \| j \rangle$(encoding the physics, independent of $m, m', q$).

This is extraordinarily powerful: instead of computing $(2j+1)(2k+1)(2j'+1)$ matrix elements, symmetry reduces the problem to a single reduced matrix element. Selection rules follow automatically:$m' = m + q$ and the triangle rule $|j - k| \leq j' \leq j + k$.

Example: Electric dipole transitions. The position operator $\hat{r}$ is a rank-1 tensor operator. The Wigner-Eckart theorem immediately gives:

• Selection rule: $\Delta j = 0, \pm 1$ (but not $j = 0 \to j' = 0$)
• Selection rule: $\Delta m = 0, \pm 1$
• The relative intensities of transitions within a multiplet are fixed by CG coefficients
• Only the reduced matrix element $\langle j' \| r \| j \rangle$ requires dynamical calculation

Generalization to SU(3). The Wigner-Eckart theorem extends to any compact Lie group. For SU(3), tensor operators transform under specific representations, and matrix elements factorize into SU(3) Clebsch-Gordan coefficients (isoscalar factors) times reduced matrix elements. This is the foundation of the Gell-Mann-Okubo mass formula relating hadron masses within SU(3) flavor multiplets.

10. Numerical Exploration: Representations with Python

This simulation constructs SU(2) representations, computes Clebsch-Gordan coefficients, verifies orthogonality of characters, and explores the SU(3) decomposition.