---
tag: 9V6PRD1Z90
kind: example
name: Representation theory of SU(2)
msc: ["22E70", "22E45", 22C05]
illustrates: [7JA7WH92K0]
uses: [E5P8AG6CK7, 7JA7WH92K0, FX3NEFYFFN, 2C3PAQ7H50]
sources:
  - url: https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)
same_as:
  wikidata: Q7314224
  wikipedia: Representation theory of SU(2)
---

## Example

Let $G = \mathrm{SU}(2)$ be the [[group|E5P8AG6CK7]] of complex $2 \times 2$ unitary matrices of determinant $1$.
For each integer $n \ge 0$, let $V_n$ be the space of [[homogeneous polynomials|2C3PAQ7H50]] of degree $n$ in two complex variables $z = (z_1, z_2)$, of dimension $n + 1$.
Put $(\pi_n(g) P)(z) = P(g^{-1} z)$ for $g \in G$ and $P \in V_n$.
This is a continuous homomorphism $G \to \mathrm{GL}(V_n)$, so it is a [[representation|7JA7WH92K0]] of $G$.

The classification: each $V_n$ is [[irreducible|FX3NEFYFFN]], the $V_n$ are pairwise non-isomorphic, and each irreducible finite-dimensional complex representation of $G$ is isomorphic to exactly one $V_n$.

Check of the character on $t_\theta = \operatorname{diag}(e^{i\theta}, e^{-i\theta})$, with $\theta \notin \pi\Z$.
The monomial $z_1^a z_2^b$ with $a + b = n$ is an eigenvector with eigenvalue $e^{i(b - a)\theta}$, so
$$\tr \pi_n(t_\theta) = \sum_{j=0}^{n} e^{i(n - 2j)\theta} = \frac{\sin\big((n+1)\theta\big)}{\sin \theta}.$$
For $n = 1$ this gives $2 \cos \theta$, the trace of $t_\theta$ itself.
Since each element of $G$ is conjugate to some $t_\theta$, these characters determine the representations.

## Remarks

Tensor products decompose by the Clebsch–Gordan rule: $V_m \otimes V_n \cong V_{m+n} \oplus V_{m+n-2} \oplus \cdots \oplus V_{\abs{m-n}}$.
The element $-I$ acts on $V_n$ by $(-1)^n$, so $V_n$ is a representation of $\mathrm{SO}(3) = \mathrm{SU}(2)/\set{\pm I}$ exactly when $n$ is even.
In physics $V_n$ is the spin $n/2$ representation; odd $n$ gives the half-integer spins of electrons and other fermions.
On the Lie algebra, $V_n$ is the irreducible representation of $\mathfrak{sl}_2(\C)$ of highest weight $n$.

## Explanations

### L1 human

The group $\mathrm{SU}(2)$ is the group of rotations of quantum spin.
Its basic ways of acting on vector spaces are labelled by one number $n = 0, 1, 2, \dots$, and the $n$-th one has dimension $n + 1$.
They are built from polynomials of degree $n$ in two variables.
Every other finite-dimensional representation is a sum of these basic ones.