A natural question covering both this and this 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:**

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).

For $\lambda=(m)$, we have $s_\lambda=h_m$, and so one of the questions linked above shows that $s_\lambda$ is irreducible.

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.