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.
One way of saying this is that the category of representations of the symmetric group (in characteristic 0) has a categorified sl(infinity) action, actually obtained by decomposing the functors of induction and restriction according to the eigenvalues of the Jucys-Murphy element. This can be turned into a categorification of sl(n)-rep by only considering representations with fixed JM eigenvalues.
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).