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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 element $X_n$ which commutes with $S_{n-1}$. The different $S_{n-1}$ representations are the different eigenspaces of the J-M 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 which I think might be relevant to your question.

EDIT (3/16/11): When I first read this question, I thought "hmmm, Weyl algebra? Really? I feel like I never hear people say they're going to categorify the Weyl algebra, but it looks like that's what the question is about..." Now I understand what's going on. Not to knock the OP, but there's a much bigger structure here he left out. If you have any $S_n$ representation $M$, you get a functor $$\operatorname{Ind}_{S_m\times S_n}^{S_{m+n}}-\otimes M: S_m-\operatorname{rep}\to S_{m+n}-\operatorname{rep}$$ and these functors all have adjoints which I won't bother writing down. All of these together categorify a Heisenberg algebra, which is what Khovanov proves in the paper linked below (though cruder versions of these results on the level the OP was talking about are much older, at least as far back as Leclerc and Thibon).

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 element $X_n$ which commutes with $S_{n-1}$. The different $S_{n-1}$ representations are the different eigenspaces of the J-M 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 which I think might be relevant to your question.

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 element X_nwhich commutes with S_{n-1}. The different S_{n-1} representations are the different eigenspaces of the J-M 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 http://latex.mathoverflow.net/png?F%5Fm%3D%28k%5BS%5Fn%5D%2F%28X%5Fn%2Dm%29%29%0A%5Cotimes%5F%7Bk%5BS%5F%7Bn%2D1%7D%5D%7D%20V.

These functors E_m,F_m satsify the relations of the Serre relations for \mathfrak{sl}(\infty) http://latex.mathoverflow.net/png?%5Cmathfrak%7Bsl%7D%28%5Cinfty%29. Over characteristic 0, these are all different, and you can think of this as an \mathfrak{sl}(\infty) http://latex.mathoverflow.net/png?%5Cmathfrak%7Bsl%7D%28%5Cinfty%29. If instead, you take representations over characteristic p, ythen 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) http://latex.mathoverflow.net/png?%5Cwidehat%7B%5Cmathfrak%7Bsl%7D%7D%28p%29.

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 which I think might be relevant to your question.

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 element $X_n$ which commutes with $S_{n-1}$. The different $S_{n-1}$ representations are the different eigenspaces of the J-M 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 which I think might be relevant to your question.

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 element X_nwhich commutes with S_{n-1}. The different S_{n-1} representations are the different eigenspaces of the J-M 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 http://latex.mathoverflow.net/png?F%5Fm%3D%28k%5BS%5Fn%5D%2F%28X%5Fn%2Dm%29%29%0A%5Cotimes%5F%7Bk%5BS%5F%7Bn%2D1%7D%5D%7D%20V.

These functors E_m,F_m satsify the relations of the Serre relations for \mathfrak{sl}(\infty) http://latex.mathoverflow.net/png?%5Cmathfrak%7Bsl%7D%28%5Cinfty%29. Over characteristic 0, these are all different, and you can think of this as an \mathfrak{sl}(\infty) http://latex.mathoverflow.net/png?%5Cmathfrak%7Bsl%7D%28%5Cinfty%29. If instead, you take representations over characteristic p, ythen 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) http://latex.mathoverflow.net/png?%5Cwidehat%7B%5Cmathfrak%7Bsl%7D%7D%28p%29.

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).

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 element X_nwhich commutes with S_{n-1}. The different S_{n-1} representations are the different eigenspaces of the J-M 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 http://latex.mathoverflow.net/png?F%5Fm%3D%28k%5BS%5Fn%5D%2F%28X%5Fn%2Dm%29%29%0A%5Cotimes%5F%7Bk%5BS%5F%7Bn%2D1%7D%5D%7D%20V.

These functors E_m,F_m satsify the relations of the Serre relations for \mathfrak{sl}(\infty) http://latex.mathoverflow.net/png?%5Cmathfrak%7Bsl%7D%28%5Cinfty%29. Over characteristic 0, these are all different, and you can think of this as an \mathfrak{sl}(\infty) http://latex.mathoverflow.net/png?%5Cmathfrak%7Bsl%7D%28%5Cinfty%29. If instead, you take representations over characteristic p, ythen 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) http://latex.mathoverflow.net/png?%5Cwidehat%7B%5Cmathfrak%7Bsl%7D%7D%28p%29.

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 which I think might be relevant to your question.