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Let L be the lattice of Young diagrams ordered by inclusion and let Ln denote the nth rank, i.e. the Young diagrams of size n. Say that lambda > mu if lambda covers mu, i.e. mu can be obtained from lambda by removing one box and let C[L] be the free vector space on L. The operators

U lambda = summu > lambda mu

D lambda = sumlambda > mu mu

are a decategorification of the induction and restriction operators on the symmetric groups, and (as observed by Stanley and generalized in the theory of differential posets) they have the property that DU - UD = I; in other words, Young's lattice along with U, D form a representation of the Weyl algebra.

Is this a manifestation of a more general phenomenon? What's the relationship between differential operators and the representation theory of the symmetric group?

Edit: Maybe I should ask a more precise question, based on the comment below. As I understand it, in the language of Coxeter groups the symmetric groups are "type A," so the Weyl algebra can be thought of as being associated to type A phenomena. What's the analogue of the Weyl algebra for types B, C, etc.?

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In type B, the poset of ordered pairs of Young diagrams describes the branching rules for the hyperoctahedral group. Here, the corresponding relation is DU - UD = 2I. Is there a natural way to rescale one of the variables so that these are "differential operators" in a sensible way? –  Sammy Black Oct 23 '09 at 18:31
    
If I'm not mistaken you can take U = x - dx, D = x + dx, but I don't know if this is the "natural" interpretation. –  Qiaochu Yuan Oct 23 '09 at 18:38
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3 Answers

up vote 9 down vote accepted

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.

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In the OP's defense, nothing is really lost by only considering the Weyl algebra since the basic representation of affine sl_p is irreducible on restriction to the Heisenberg subalgebra. –  David Hill Mar 16 '11 at 23:12
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Sasha Kleshchev's book "Linear and Projective Representations of Symmetric Groups" is the reference I'd suggest. Chapter 1 contains the connection with Young's lattice, and the subsequent chapters develop the functors that Ben described above. The second half of the book develops the theory for spin representations of symmetric groups which is an honest type B analogue (The functors $E_m$ and $F_m$ in Ben's answer satisfy the Serre relations for the Kac-Moody algebra of type $B_\infty$).

To add a little more detail to Ben's answer, the right level of generality to think about these questions is the affine Hecke algebra (either the degenerate or nondegenerate varieties). I'll describe the degenerate case:

Let $F$ be an algebraically closed field of characteristic $p$. As a vector space the (degenerate) affine Hecke algebra is a tensor product of a polynomial algebra with the group algebra of the symmetric group: $H_d=F[x_1,\ldots,x_d]\otimes FS_d$. Multiplication is defined so that each tensor summand is a subalgebra, and $H_d$ satisfies the mixed relations $s_ix_j=x_js_i$ for $j\neq i,i+1$ (here $s_i=(i,i+1)$), and $s_ix_i=x_{i+1}s_1-1$.

Note that in addition to being a subalgera, $FS_d$ is also a quotient of $H_d$ obtained by mapping $s_i\mapsto s_i$ and $x_1\mapsto 0$ (so that the $x_i$ map to Jucys-Murphy elements).

The polynomial subalgebra forms a maximal commutative subalgebra, so given a finite dimensional $H_d$-module $M$, we may decompose $$M=\bigoplus_{(a_1,\ldots,a_d)\in F^d}M_{(a_1,\ldots,a_d)},$$ where $$M_{(a_1,\ldots a_d)}=\lbrace m\in M|(x_i-a_i)^Nm=0,\mbox{ for }N\gg0\mbox{ and }i=1,\ldots,d \rbrace$$ is the generalized $(a_1,\ldots,a_d)$-eigenspace for the action of $x_1,\ldots,x_d$. Let $I=\mathbb{Z}1_F\subset F$ and $Rep_IH_d$ be the category of finite dimensional $H_d$-modules which are integral in the sense that if $M\in Rep_IH_d$, and $M_{(a_1,\ldots,a_d)}\neq 0$, then $(a_1,\ldots,a_d)\in I^d$.

Now, let $K_d=K_0(Rep_IH_d)$, and $K=\bigoplus_d K_d$. Then, the categorification statement is that parabolic induction and restriction give $K$ the structure of a bialgebra, and as such $$K\cong U_{\mathbb{Z}}(\mathfrak{n}).$$ In the above statement, $\mathfrak{n}\subset \mathfrak{g}$ is the maximal nilpotent subalgebra of the Kac-Moody algebra $\mathfrak{g}$ generated by the Chevalley generators $e_i$, where, if $char F=p$, then $\mathfrak{g}=\hat{sl}(p)$, and if $char F=0$, $\mathfrak{g}=\mathfrak{gl}(\infty)$. In both cases $U_{\mathbb{Z}}(\mathfrak{g})$ denotes the Kostant-Tits $\mathbb{Z}$-subalgebra of the universal enveloping algebra. Note here that the Chevalley generators are indexed by $I$.

Now, for each dominant integral weight $\Lambda=\sum_{i\in I}\lambda_i\Lambda_i$ ($\Lambda_i$ the fundamental dominant weights) for $\mathfrak{g}$, define the polynomial $$f_\Lambda=\prod_{i\in I}(x_1-i)^{\lambda_i}.$$ Then, the algebra $H_d^\Lambda=H_d/(H_d f_\Lambda H_d)$ is finite dimensional. In the case $\Lambda=\Lambda_0$, $H_d^\Lambda\cong FS_d$.

One can form $K_d(\Lambda)$ and $K(\Lambda)$ as above corresponding to the category $H_d^\Lambda-mod$. Then, the categorification statement is $$K(\Lambda)\cong V_{\mathbb{Z}}(\Lambda)$$ as $\mathfrak{g}$-modules, where $V(\Lambda)$ is the irreducible $\mathfrak{g}$-module of highest weight $\Lambda$ generated by a highest weight vector $v_+$, and $V_{\mathbb{Z}}(\Lambda)=U_\mathbb{Z}(\mathfrak{g})v_+$ is an admissible lattice. The action of the Chevalley generators on $K$ are analogues of the functors in Ben's answer. The action of the Weyl module corresponds to the action of $D=\sum_{i\in I}e_i$ and $U=\sum_{i\in I}f_i$ (in characteristic 0, this is defined in the completion $\mathfrak{a}(\infty)$ of $\mathfrak{gl}(\infty)$.

One can generalize this story to $\hat{\mathfrak{sl}}_\ell$ by working with the (nondegenerate) affine Hecke algebra $H_d(t)$, where $t$ is a primitive $\ell$-th root of unity. In this case, the finite dimensional quotients are Hecke algebras of complex reflection groups. The hyperoctohedral group corresponds to the highest weight $\Lambda=2\Lambda_0$. Then $V(\Lambda)$ is a level 2 representation, hence the central element acts by $2\cdot Id$ as in Sammy's comment).

In the second half of Kleshchev's book, the Hecke algebra is replaced by the so-called Hecke-Clifford (or Sergeev) algebra, and $\mathfrak{g}$ is of type $B_\infty$ or $A_{2\ell}^{(2)}$ depending on the ground field (or one can work in the non-degenerate case so that $\ell$ needn't be prime.

The algebras introduced by Khovanov-Lauda and Rouquier generalize this story to arbitrary symmetrizable Kac-Moody algebra. These algebras are graded, so one gets a categorification of the quantum group $U_q$, where $q$ keeps track of the grading . . .

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Another general perspective on this phenomenon is one described in the paper "Combinatorial Hopf algebras and towers of algebras - dimension, quantization and functoriality" by Bergeron, Lam and Li. There is a relation between towers of associative algebras satisfying some conditions, pairs of dual combinatorial Hopf algebras, and dual graded graphs.

Passing from towers of algebras to graded dual Hopf algebras is done by using induction and restriction on the Grothendieck groups, and then using structure constants one gets edge multiplicities for a graded graph. Dual graded graphs are generalizations of differential posets (the definig rule is $D_{\Gamma} U_{\Gamma'}-U_{\Gamma}D_{\Gamma'}=r Id$).

This generalizes the way one can start with the tower of symmetric group algebras $\bigoplus_{n\geq 0}\mathbb{C} \mathfrak{S}_n$ and get the ring of symmetric functions which is a self dual graded Hopf algebra (construction of the Hopf structure in terms of the Grothendieck groups by Zelevinsky) and finally obtain the Young lattice through Pieri rules on the ring of symmetric functions (equivalently the branching rules for the symmetric group).

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