# Why do partitions correspond to irreps in $S_n$?

As stated for example in these notes (Link to pdf), top of page 8, irreps of the symmetric group $$S_n$$ correspond to partitions of $$n$$. This is justified with the following statement:

Irreps of $$S_n$$ correspond to partitions of $$n$$. We've seen that conjugacy classes of $$S_n$$ are defined by cycle type, and cycle types correspond to partitions. Therefore partitions correspond to conjugacy classes, which correspond to irreps.

I understand the equivalence between partitions, cycle types, and conjugacy classes, but I do not fully get the connection with irreps:

1. I can associate to a partition $$\lambda\vdash n$$ the conjugacy class of permutations of the form $$\pi=(a_1,...,a_{\lambda_1})(b_1,...,b_{\lambda_2})\cdots (c_1,...,c_{\lambda_k}).$$

2. The fact that conjugacy classes are defined by cycle types comes from the fact that $$\sigma\pi\sigma^{-1}$$ has the same cycle type structure as $$\pi$$.

However, in what sense do conjugacy classes correspond to irreps? I can understand this if we restrict to one-dimensional representations, as then $$\rho(\pi)=\rho(\sigma\pi\sigma^{-1})$$ for all $$\sigma$$, but this is not the case for higher dimensional representations I think, being $$S_n$$ non-abelian.

• It is in fact a somewhat non-trivial fact that a finite group has the same number of irreducible complex representations and conjugacy classes. But it is covered in pretty much any introductory treatment of the topic. Jul 1 '20 at 17:38

In general, there is no natural bijection between conjugacy classes and irreducible representations. A nice, albeit a bit more advanced, discussion of this can be found here on MO. In the particular case of $$S_n$$, however, there is a natural choice of bijection between irreducible representations and conjugacy classes. Establishing this bijection will be part of the notes you are reading.
• This being said, it turns out that for $S_n$ the bijection is canonical, and is given by the whole theory of Young tableaux and such. Jul 1 '20 at 17:41