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?



Symmetric Powers, Tableau and Wreath Products
You want to read [Splitting the square of a Schur function into its symmetric and antisymmetric parts](http://www.ams.org/mathscinet-getitem?mr=1331743), where this question is answered in terms of domino tableaux. I am also a big fan of [Domino tableaux, Schützenberger involution, and the symmetric group action](http://www.ams.org/mathscinet-getitem?mr=1798322) which, to my mind, gives the "right" formulation of the domino tableaux rule.


