Let S_n be the group of permutations of n elements. Consider map S_n -> S_mn of block permutations, and an irreducible representation of S_mn (over complex numbers), corresponding to Young diagram Y. Naturally, it decomposes into direct sum of irreducible representations for S_n. Is it possible to give a formula for the decomposition?

Block permutations: Consider S_n as matrices in GL(n), embed GL(n) into GL(mn) as block matrices with scalar blocks of size m * m, then you got the embedding S_n -> S_mn

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