Given a Young tableau, we can construct its Young symmetrizer $c_\lambda$. Then, the ideal $\mathbb{C} S_n \cdot c_\lambda$ is an irreducible representation of $S_n$. Exercise 4.5 in Fulton and Harris asks the reader to find which Young diagrams correspond to which irreps for $S_3, S_4$, and $S_5$. It's easy for $S_3$: the diagram $(1, ..., 1)$ always corresponds to the sign representation, the diagram $(n)$ always corresponds to the trivial representation, so the diagram $(2, 1)$ must correspond to the remaining representation.
How do I connect diagrams to irreps for $S_4$ and $S_5$? I don't know what I'm really supposed to use here beside the characters of the irreps. The representations of $S_4$ and $S_5$ have previously been constructed geometrically.
EDIT: It seems that I failed to communicate my question well, so I'll try to clarify. For $S_5$, I know that there are 7 irreps corresponding to different Young diagrams by means of the Young symmetrizer. The question is, how do I learn any of their properties, such as dimensions or characters? Since one can identify the irreps of $S_5$ by other means, one way would to connect the known irreps to the Young diagrams. For example, if we make a 5-dimensional representation of $S_5$ by permuting the basis vectors of $\mathbb{C}^5$, this representation is a sum of two irreps: trivial and a four-dimensional representation $V$. This representation corresponds to some Young diagram, but I don't see a way to find which one.