For $n\geq 7$, I would like to show that $S_n$ has no irredicuble representations of dimension $m$ for $2\leq m\leq n-2$.
The catch is that I am not allowed to use any "machinery" (evidently, this problem should be solvable having knowledge only of chapter 1 of Fulton and Harris).
At the moment, I don't really have any idea of how to approach this, especially without the use of character theory. Any suggestions would be greatly appreciated!