A natural question covering both [this](https://mathoverflow.net/questions/98043) and [this](https://mathoverflow.net/questions/98474/is-an-elementary-symmetric-polynomial-an-irreducibile-element-in-the-polynomial-r) question would be 


*Let $n>2$. Describe Young diagrams $\lambda$ with at most $n$ nonempty rows (or equivalently non-increasing sequences $\lambda=(\lambda\_1\ge\lambda\_2\ge\ldots\ge\lambda\_n\ge0$) for which the Schur polynomial $$s\_\lambda(x\_1,\ldots,x\_n):=\frac{\det(x\_i^{\lambda\_j+n-j})}{\det(x\_i^{n-j})}$$ is irreducible in $\mathbb{C}[x\_1,\ldots,x\_n]$.*


**Partial results:** 


1. Of course, for $\lambda\_n>0$, this polynomial is divisible by $s\_{1^n}=e\_n$ , and so isn't irreducible. (This is a generalisation of the fact that $e\_k=s\_{1^k}$ can only be irreducible for $k< n$ (which is shown to be true in an answer to one of the questions linked above).
2. For $\lambda=(m)$, we have $s\_\lambda=h\_m$, and so one of the questions linked above shows that $s\_\lambda$ is irreducible.
3. For $\lambda=(n-1,n-2,\ldots,0)$ we have $s\_\lambda=\prod\_{i< j}(x\_i+x\_j)$ by Vandermonde, which is *very* reducible.


Of course, it is mere curiousity forcing me to ask it, but I think that this question has a potential to have a meaningful answer.



Irreducibility of Schur polynomials
Let $g=\gcd(\lambda\_1+n-1,\lambda\_2+n-2,\dots,\lambda\_n)$. And let $V \_g(X)=\prod \_{i < j}\frac{x\_i^g-x\_j^g}{x\_i-x\_j}$. David already mentioned that $s\_{\lambda}(X)$ is divisible by $V\_g(X)$. This is in fact all that can happen.


**Theorem:** For every partition $\lambda$ (with $\lambda\_n=0$), the polynomial $\frac{s\_{\lambda}(X)}{V\_g(X)}$ is either constant or irreducible.


This follows from theorem 3.1 in the paper ["Newton functions generating symmetric fields and irreducibility of Schur polynomials"](http://www.sciencedirect.com/science/article/pii/S0001870809002059) by Dvornicich and Zannier. 


A result which is somewhat more general is stated and proved in ["On the irreducibility of irreducible characters of simple Lie algebras"](http://arxiv.org/abs/1110.5300). They extend the irreducibility above to characters of irreducible representations of simple Lie algebras divided by a Weyl denominator type factor (analog of $V\_g$ above). Schur functions are a special case corresponding to $GL\_n$.


