# Symmetric Powers, Tableau and Wreath Products

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?

• David's provided exactly the answer I wanted (a win for MathOverflow!). I've edited the question (removed a confusing wrong bit) so it should be clearer for anyone else interested. – James Griffin Nov 18 '09 at 10:15