Topic 4 of 8

Representation Theory

A representation assigns a matrix to each group element in a way that respects the group operation. This is how symmetry groups act on physical states. Every quantum number — spin, isospin, color charge — labels an irreducible representation. Schur's lemma, character theory, and the sum-of-squares formula are the essential tools.

Key Concepts

Representation

A homomorphism

\rho: G \to GL(V)

assigning an invertible linear map on a vector space

V

to each

g \in G

, with

\rho(gh) = \rho(g)\rho(h)

Dimension

\dim(V)

, the size of the matrices. Physical states transforming under an

n

-dimensional representation live in an

n

-dimensional space.

Irreducible Representation (irrep)

A representation with no proper

G

-invariant subspace. Every representation decomposes into irreps (complete reducibility for compact groups). Particle types correspond to irreps.

Character

\chi(g) = \text{Tr}(\rho(g))

. Determines the representation up to isomorphism. Characters of non-conjugate irreps are orthogonal.

Schur's Lemma

Any intertwining operator between two irreps is either zero or an isomorphism. Consequence: in an irrep of an abelian group, all matrices are

\lambda I

— abelian group irreps are 1-dimensional.

Key Equations

Sum of Squares (Burnside)

|G| = \sum_i (\dim V_i)^2

Sum over all irreps

V_i

. The number of irreps equals the number of conjugacy classes.

Character Orthogonality

\frac{1}{|G|}\sum_{g \in G} \chi_i(g)^*\, \chi_j(g) = \delta_{ij}

Characters of distinct irreps are orthogonal under the group inner product.

Dimension of

SU(2)

Irreps

\dim V_j = 2j + 1, \qquad j = 0,\,\tfrac{1}{2},\,1,\,\tfrac{3}{2},\,\ldots

Each non-negative half-integer

j

labels a unique irrep of

SU(2)

Worked Example

Irrep Dimensions of $S_3$

Problem

$S_3$ has order 6 and 3 conjugacy classes. The irrep dimensions satisfy $\sum_i d_i^2 = 6$ . Two irreps are 1-dimensional. Find the dimension of the third.

Solution

Number of irreps = number of conjugacy classes = 3.

Let the dimensions be $d_1=1$ , $d_2=1$ , $d_3=?$ . Then $1^2+1^2+d_3^2=6$ .

$d_3^2 = 4$ , so $d_3 = 2$ .

The three irreps of $S_3$ are: trivial (dim 1), sign (dim 1), and standard (dim 2).

Answer

d_3 = 2

Practice

Exercises

6 problems

1 of 6

For $SU(2)$ , the spin- $j$ irrep has dimension $2j+1$ . What is the dimension of the $j=5/2$ representation?

Your answer

2 of 6

The Casimir invariant for the spin- $j$ representation of $SU(2)$ has eigenvalue $j(j+1)$ . For $j=2$ , what is this value?

Your answer

3 of 6

$S_3$ has order 6 and its irrep dimensions satisfy $\sum d_i^2 = 6$ . If two irreps have dimension 1, what is the dimension $d$ of the third irrep?

Unlock Exercise 3

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →

4 of 6

The number of irreducible representations of a finite group equals the number of conjugacy classes. $S_4$ has 5 conjugacy classes. How many irreps does $S_4$ have?

Unlock Exercise 4

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →

5 of 6

The Casimir eigenvalue for $j=3/2$ is $j(j+1) = (3/2)(5/2)$ . What is this value? Enter as a decimal.

Unlock Exercise 5

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →

6 of 6

The adjoint representation of a Lie group has dimension equal to the number of generators $= n^2-1$ for $SU(n)$ . What is the dimension of the adjoint of $SU(3)$ ?

Unlock Exercise 6

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →

Key Takeaways

A representation $\rho: G\to GL(V)$ lets the group act on a vector space. Physical states transforming under $G$ form a representation.
Every representation of a compact group decomposes uniquely into irreps. Particle species correspond to specific irreps.
For $SU(2)$ , irreps are labeled by spin $j = 0, \frac{1}{2}, 1, \ldots$ , with dimension $2j+1$ and Casimir eigenvalue $j(j+1)$ .
The number of irreps equals the number of conjugacy classes; their squared dimensions sum to $|G|$ (Burnside's theorem).

Representation Theory

Key Concepts

Key Equations

Irrep Dimensions of S3S_3S3​

Exercises

Key Takeaways

Irrep Dimensions of $S_3$