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Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams.

I have seen the definition of plethysms in symmetric functions. I would like to understand the corresponding realisation of plethysms as the regular representations of the permutation group, in this case $S_{mn}$.

If possible can you please explain this in a way a physicist can understand?

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    $\begingroup$ For the symmetric group, the representation theoretic significance of plethysm is related to wreath products. See Theorem A2.8 of Stanley's Enumerative Combinatorics, Vol 2. (By the way, plethysm has a different representation-theoretic significance for the general linear group; and that one is probably a bit easier to understand- it is composition of characters). $\endgroup$ Jun 17, 2015 at 14:51
  • $\begingroup$ @SamHopkins Thanks for your reference. I have difficulty trying to understand wreath products, can you perhaps give an elementary example how this works or some reference? $\endgroup$
    – vishmay
    Jun 17, 2015 at 15:25
  • $\begingroup$ Example A2.9 in EC2 is a good example, imo. $\endgroup$ Jun 17, 2015 at 15:38

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