The systematic method for the explicit construction of representations of su(2) algebra? (e.g. pauli matrices)

Question

How do I construct the su(2) representations of a given dimension?

Answer 1

There are many ways but it's a rather lenghty derivation if you want to find all representations. I don't think people here will serve you the whole derivation. And I'm talking about finite dimensional representations here.

The usual way is to reformulate the algebra with raising and lowering operators and use these to construct the representation of the desired dimension. This is done by loads and loads of (computationally simple) eigenvector business.

You will find all the answer you're looking for in standard textbooks, for example An Elementary Introduction to Groups and Representations by Brian Hall.

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Anyway, here is a direct path to construct a complex representation of any dimension:

You get the Lie Algebra $\text{su}(2)$ as the tangent space of the Lie Group $SU(2)$ at the unit element. How to get the $m$-dimensional, irreducible representations? You know the fundamental, two dimensional representation acting on vectors $z=(z_1,z_2)$, i.e. the set of unitarily matrices $U$ with complex entries and determinant $1$.

Now consider the polynomials of the form $$p_{m+1}(z)\equiv p_{m+1}(z_1,z_2)=a_0 z_1^m+a_1z_1^{m-1}z_2+a_2z_1^{m-2}z_2^2+ \cdots+ a_{m-1}z_1z_2^{m-1}+a_{m}z_2^{m},$$ viewed as vector space with elements $a=(a_0,a_1, \ldots, a_m)$, then $$\Pi_{m+1}(U):p_{m+1}(z)\longrightarrow p_{m+1}(U^{-1}z),$$ is an $m+1$-dimensional representation.

You can sit down with pen and paper, choose a small $m$ and watch how $U^{-1}$ messes up the coefficients of the polynomial (i.e. maps to another vector) for yourself. Consider a set of $a$-basis vectors and you have your $m+1$ dimensional $\Pi_{m+1}(U)$ matrix. Now express $U$ it terms of the three angles ($SU(3)$ is a three dimensional manifold), compute the derivatives in all directions, set the angles to zero and you have your Lie algebra basis.

You also find odd dimensional representations by considering representations of $SO(3)$, so you might wanna study the behaviour of subsets of spherical hermonics under rotation.