There is a much more general story here, though one my brain is not very up to explaining it this afternoon, and unfortunately, I don't know of anywhere it's summarized well for beginners.

So, how you you prove the restriction rule you mentioned above?  You note that the restriction of a $S_n$ rep to an $S_{n-1}$ rep has an action of the [Jucys-Murphy][1] element $X_n$ which commutes with $S_{n-1}$.  The different $S_{n-1}$ representations are the different eigenspaces of the J-M element.

[1]:http://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element

So, one can think of "restrict and take the m-eigenspace" as a functor $E_m$; this defines a direct sum decomposition of the functor of restriction.

Of course, this functor has an adjoint: I think the best way to think about this is as $F\_m=(k[S\_n]/(X\_n-m))
\otimes\_{k[S\_{n-1}]} V$.

These functors `E_m,F_m` satsify the relations of the Serre relations for  $\mathfrak{sl}(\infty)$. Over characteristic 0, these are all different, and you can think of this as an $\mathfrak{sl}(\infty)$.  If instead, you take representations over characteristic p, then `E_m=E_{m+p}` so you can think of them as being in a circle, an affine Dynkin diagram, so one gets an action of $\widehat{\mathfrak{sl}}(p)$.

Similar categorifications of other representations can deconstructed in general by looking at representations of complex reflection groups given by the wreath product of the symmetric group with a cyclic group.  So, Sammy, you shouldn't rescale, you should celebrate that you found a representation with a different highest weight (also, if you really care, you should go talk to Jon Brundan or Sasha Kleshchev; they are some of the world's experts on this stuff).

**EDIT:** Khovanov has actually just posted [a paper](http://arxiv.org/abs/1009.3295) which I think might be relevant to your question.