Let $S_n$ be the group of permutations of $n$ elements. Consider the map $S_n \to S_{mn}$ of block permutations, and an irreducible representation of $S_{mn}$ (over the complex numbers), corresponding to a Young diagram $Y$. Naturally, it decomposes into a direct sum of irreducible representations for $S_n$. Is it possible to give a formula for this decomposition?
Block permutations: Consider $S_n$ as matrices in $\mathrm{GL}(n)$, embed $\mathrm{GL}(n)$ into $\mathrm{GL}(mn)$ as block matrices with scalar blocks of size $m \times m$, then you got the embedding $S_n \to S_{mn}$.