Irreducible representations of $su2$ in dimension $d$ : how do we know it exists?

I am studying in a book the way to find the irreducible representations of $su(2)$.

They start by assuming that it exists a basis of the representation in a vectorial space of dimension $d$ ($d$ is any).

And they do all the work after it.

But it all started with the assumption that such a representation exist for any dimension $d$.

How do we know that ?

1 Answer

When you finally arrive at an understanding of how a $d$-dimensional irreducible representation should look like, you've identified it's basis and how $\mathfrak{su}(2)$ acts on this basis. Now we can safely turn things around: take a $d$-dimensional vector space and define the action of $\mathfrak{su}(2)$ on it's basis by the rules that you've derived, and you get this representation.

Alternatively, you can take the defining 2-dimensional representation $V$ of $\mathfrak{su}(2)$, and take it's symmetric powers $S^d V$, which appear to be $d+1$-dimensional and irreducible, proving the existence. This has the advantage that corresponding irreducible representations of the group $SU(2)$ can also be found the same way, using symmetric powers of the defining representation.