TITLE: Symmetric Powers, Tableau and Wreath Products
QUESTION [7 upvotes]: Let V and W be irreducible representations of $S_n$ and $S_m$ over a field of characteristic 0.  Then the Littlewood-Richardson coefficients allow us to compute the isomorphism type of the induced $S_{n+m}$-module V⊗W↑.  This induction comes from the inclusion
$S_n\times S_m \rightarrow S_{n+m}$.
Now suppose V=W.  Then V⊗V↑ is a $S_{2n}$-module.  But actually there's a symmetry coming from the symmetric monoidal category structure, so there is another induction up to an $S_{2n}$-module structure:
Extend the action of $S_n\times S_n$ on V⊗V by including the symmetry $c_V$, this naturally extends the group to the wreath product $S_n\sim S_2$.  Induction along
$S_n\sim S_2 \rightarrow S_{2n}$
gives the representation that I want:
$(V\otimes V)_{S_n\sim S_2}\uparrow^{S_{2n}} \hookrightarrow V\otimes V\uparrow^{S_{2n}}$.
Using the Littlewood-Richardson rules we know the structure of the last term in terms of semi-standard skew tableau.  My question is, how do we characterise the inclusion?

REPLY [2 votes]: My understanding was that induction from wreath products was supposed to correspond to plethystic substitution of symmetric functions.  But I don't really understand that, so I can't really say.