# How do Specht modules give us the irreducible representations of $S_n$?

I want to understand the Specht module and use it to find the irreducible representations of $S_n$. I know that the Specht modules are spanned by polytabloids, which are constructed using the permutations $\sigma \in S_n$ belonging to the same conjugacy class.

My understanding is that since the Specht modules are cyclic (generated by a finite number of polytabloids), all one has to do to figure out the corresponding irreducible representation of $S_n$ is observe how an arbitrary permutation $\sigma \in S_n$ acts on each of the generators of the Specht module. Is this correct?

Is there a way of knowing how many generators there are (and what these generators are) for a Specht module corresponding to a general cycle shape $\lambda$? Thank you.

• You know that irreps or S_n are in bijection with conjugacy classes so are in bijection with partitions and so are bijection with {Specht modules}. So the only thing you are left to show is that each Specht module is irreducible. This is standard material, but I did a (hopefully clearer than standard) writeup on this some time ago; see questions 15-16 from math.columbia.edu/~maithreya/MLQFTGMvol1.pdf (q 15 answers your question and q 16 proves the result that the answer cites) – Maithreya Sitaraman Aug 2 '18 at 18:29