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.