It is a basic fact about the symmetric group $S\_n$ that its irreducible representations are indexed by partitions of $n$.


My question is, can the association between partitions and irreps be specified without having to explicitly construct the irreps? In other words, is there for each partition $\lambda$ some relatively simple property that is possessed by the Specht module $S^\lambda$ but none of the other irreps of $S\_n$?


In particular I was imagining that an answer might come from Jucys-Murphy type theory but I am not familiar enough with the material to know how to (or if it is possible to) use it to characterize $S^\lambda$ without having to give a bunch of background information.



Is there a non-explicit characterization of the Specht modules?
The characterization using Jucys-Murphy elements is this: let $z$ be a formal variable, and $X\_i$ the Jucys-Murphy elements. The product $(z-X\_1)\cdots (z-X\_n)$ is central in the group algebra of the symmetric group, so it acts on any irreducible representation by a polynomial with scalar coefficients. On the Specht module $V\_\lambda$, the polynomial it acts by the product of $z+i-j$ where $(i,j)$ are the boxes of the Young diagram. 
This is one of the consequences of the Okounkov-Vershik theory as mentioned by Steven and Andrew above.


**EDIT:** By way of normalization, note that in $S\_2$ this is $z^2-z\cdot (12)$, so it acts on the trivial rep by $z(z-1)$ and on the sign by $z(z+1)$. The difference comes from the fact that the box $(1,2)$ is in the diagram of the trivial, and $(2,1)$ is in the diagram of the sign. So, if somewhere you flipped this convention (which is very easy), you must change $z+i-j$ to $z-i+j$.


