Posted by: Steven Sam | August 31, 2009

A Fock space representation

Lately I’ve been reading about quantum groups. I particularly enjoyed Bernard Leclerc’s paper Symmetric functions and the Fock space representation of U_q(\widehat{\mathfrak{sl}_n}), so I want to discuss a little bit of it. In particular, I want to give an example of the technique of deforming an object in order to see “hidden structure.” My notation will differ slightly from Leclerc’s since he uses the French notation for Young diagrams.

One thing that has always been really hard for me to wrap my head around is the really complicated presentations that affine Lie algebras have and how one is supposed to do anything with them. This post will be about the affine Lie algebra \widehat{\mathfrak{sl}_n}, which is defined in the paper as the Lie algebra with generators e_i, f_i, h_i (0 \le i \le n-1) and d, with 5 lines of relations. Let K be a field of characteristic 0. Another way around this is to first define the loop algebra L(\mathfrak{g}) of a simple Lie algebra \mathfrak{g} as: L(\mathfrak{g}) = \mathfrak{g} \otimes K[t,t^{-1}] with a Lie bracket given by [a \otimes t^n, b \otimes t^m] = [a,b]_{\mathfrak{g}} \otimes t^{n+m}, and then to say that \widehat{\mathfrak{g}} is its universal central extension. More precisely, we say add a central element c, and then extend the bracket above via [a \otimes t^n, b \otimes t^m] = [a,b]_{\mathfrak{g}} \otimes t^{n+m} + (a,b)_{\mathfrak{g}} n \delta_{n,-m} c where \delta is the Kronecker delta, and (,) is the Killing form of \mathfrak{g}.

In the case that \mathfrak{g} is \mathfrak{sl}_n, I want to discuss a more concrete (combinatorial) description. Just as \mathfrak{sl}_n can be thought as the traceless operators on an n-dimensional vector space (the standard representation), we can also find a standard representation for \widehat{\mathfrak{sl}_n} (the Fock space representation). For this, let {\cal F} denote the ring of symmetric functions over K in infinitely many variables. The Schur functions s_\lambda form a basis indexed by partitions, and this will be our representation. In order to describe the actions of \widehat{\mathfrak{sl}_n} on Sym, we’ll need some notation.

First, we represent partitions \lambda by their Young diagram (\lambda_i boxes drawn in the ith row, left justified). The content of a box (i,j) is the number i-j. We’ll consider contents modulo n, and say that a box is an i-node if it has content i modulo n. We define e_is_\lambda = \sum_\mu s_\mu (resp. f_is_\lambda = \sum_\mu s_\mu) where the sum is over all \mu obtained from \lambda by removing (resp. adding) an i-node, and define ds_\lambda = N_0(\lambda)s_\lambda where N_0(\lambda) is the number of 0-nodes of \lambda. Finally, set h_i = e_if_i - f_ie_i. Then \widehat{\mathfrak{sl}_n} is the Lie algebra spanned by these generators.

Unlike the \mathfrak{sl}_n case, the Fock space representation is not irreducible. Let p_\lambda = \prod_i p_{\lambda_i} be the power sum symmetric function where p_i = \sum_{j \ge 1} x_j^i. It turns out that the set \{p_{n\lambda}\} where n\lambda = (n\lambda_1, n\lambda_2, \dots) are the highest weight vectors of this representation (i.e., they are killed by the e_i, and are eigenvectors for d and the h_i). Furthermore, one has dp_{n\lambda} = |\lambda| p_{n\lambda}, so we have a natural notion of degree for our highest weight vectors. Within these graded subsets, the p_{n\lambda} give an obvious choice of basis, but there is no reason to favor them: for example, \{ p_{(2n)} + p_{(n,n)}, p_{(2n)} - p_{(n,n)} \} also forms a basis for the highest weight vectors of degree 2. The point is that the Schur functions give a “natural basis” for {\cal F} in the sense that we have defined our operators in this basis, and the basis of highest weight vectors should have “nice” properties with respect to this fixed basis, although it’s not clear what nice means right now.

The next part is something that I am learning to appreciate: since there is no way to figure out a “canonical basis” for the highest weight vectors, we should introduce a new parameter to make the structure of the Fock space representation more rigid. This new parameter is made precise by replacing \widehat{\mathfrak{sl}_n} by its q-analogue U_q(\widehat{\mathfrak{sl}_n}), and similarly for {\cal F}. More precisely, we can’t deform the Lie algebra, but we can deform its universal enveloping algebra. The quantum group U_q(\widehat{\mathfrak{sl}_n}) has generators E_i, F_i, K_i, K_i^{-1}, and D, D^{-1} and even more relations than \widehat{\mathfrak{sl}_n} has, so rather than give those, I just want to mention how to change the action on {\cal F}. Let q be a transcendental element over K, and let K(q) be the function field over K. We set {\cal F}_q = {\cal F} \otimes_K K(q), and to get the actions of the E_i and F_i on {\cal F}_q, we’ll use almost the same formulas as above, but we’ll need a bit more partition notation.

Given a partition, a box is removable (resp. indent) if it can be removed (resp. added) to obtain another Young diagram. Let \lambda and \mu be two partitions such that \lambda is obtained from \mu by adding an i-node \gamma to it. Let I^r_i(\lambda, \mu) (resp. R^r_i(\lambda, \mu)) be the number of indent i-nodes of \lambda (resp. number of removable i-nodes of \lambda) which are strictly to the right of \gamma. Also set N^r_i(\lambda, \mu) = I^r_i(\lambda, \mu) - R^r_i(\lambda, \mu). Define the same numbers with the superscript r replaced by l by replacing “right” with “left.” Then we set F_is_\lambda = \sum_\mu q^{N_i^r(\lambda, \mu)} s_\mu (resp. E_is_\nu = \sum_\mu q^{-N_i^l(\mu, \nu)} s_\mu) where the sum is over all \mu such that \mu / \lambda is an i-node (resp. \mu / \nu is an i-node). We also set D^{\pm}s_\lambda = q^{\pm N_0(\lambda)}s_\lambda and define K^{\pm}_is_\lambda = q^{\pm K(i, \lambda)}s_\lambda where K(i, \lambda) is the number of removable and indent i-nodes of \lambda. And we can take the quantum group U_q(\widehat{\mathfrak{sl}_n}) to be the K(q) algebra spanned by these generators.

We’ll use a family of operators V_k to find a nice basis. To define their actions, we need some definitions about ribbons. First, an m-ribbon is a connected skew Young diagram with m boxes which does not contain a 2 \times 2 square. The most northeast box of an m-ribbon is called its origin. Its spin is the number of rows it has minus 1. A connected union of m-ribbons is a horizontal m-ribbon strip if it is a skew Young diagram, and if the origin of each ribbon does not lie below another box in the same column. The weight of a horizontal m-ribbon strip is the number of m-ribbons used to build it. Any tiling of a horizontal m-ribbon strip subject to these constraints is unique, so we can define the spin of a horizontal m-ribbon strip to be the sum of the spins of its ribbons. We define V_k s_\lambda = \sum_\mu (-q)^{{\rm spin}(\mu/\lambda)} s_\mu where the sum is over all \mu such that \mu / \lambda is a horizontal n-ribbon strip of weight k. This sort of looks like the definitions one uses to define the Murnaghan–Nakayama rule for multiplying a Schur function and power sum symmetric function. In fact, in the classical limit “q=1”, V_k reduces to multiplication by the plethysm h_n \circ p_k = h_n(x_1^k, x_2^k, \dots ).

We introduce a K-linear bar involution on K(q) via q \mapsto q^{-1}, and then extend this to a compatible K-linear involution x \mapsto \overline{x} on {\cal F}_q by requiring that it commute with the actions of F_i and V_k on {\cal F}_q, and that it fixes the basis vector s_\emptyset. Let L (resp. L^-) be the free {\bf Z}[q]-submodule (resp. {\bf Z}[q^{-1}]-submodule) of {\cal F}_q spanned by the basis \{s_\lambda\}. Then we have the following theorem.

Theorem. There exist two unique bar-invariant bases B = \{G(\lambda)\} and B^- = \{G^-(\lambda)\} of {\cal F}_q such that G(\lambda) = s_\lambda \pmod {qL} and G^-(\lambda) = s_\lambda \pmod {q^{-1}L^-}.

The two bases are called the canonical basis and dual canonical basis of {\cal F}_q. They have a lot of nice properties. Going back to highest weight vectors, it turns out that G^-(n\lambda) is a highest weight vector for all \lambda. Furthermore, “setting q=1” this basis of highest weight vectors reduces to the plethysms s_\lambda \circ p_n = s_\lambda(x_1^n, x_2^n, \dots) (this is related to the classical limit of the operators V_k). Since it comes from a more “rigid” basis, we might be satisfied with this choice for a basis of highest weight vectors in {\cal F}. Another nice property which happens with canonical bases is a nonnegativity property: write G(\mu) = \sum_\lambda d_{\lambda, \mu}(q) s_\lambda and G^-(\lambda) = \sum_\mu e_{\lambda, \mu}(-q^{-1}) s_\mu where the d and e are polynomials.

Theorem.The polynomials d and e have nonnegative coefficients as polynomials in q. Furthermore, d_{\lambda, \mu}(q) is nonzero only if \lambda \le \mu and similarly, e_{\lambda, \mu}(q) is nonzero only if \mu \le \lambda.

Here we are using the dominance order on partitions: \lambda \le \mu if \lambda - \mu can be written as a nonnegative linear combination of vectors \varepsilon_i - \varepsilon_{i+1} where \varepsilon_i is the vector with a 1 in the ith coordinate and 0s in the other coordinates.

There is a bunch of other stuff which Leclerc discusses in the paper, like connections to Kazhdan-Lusztig polynomials and Macdonald polynomials, which illustrates why these canonical bases and their change of basis matrices are important, but I’ll stop here.




  1. “One thing that has always been really hard for me to wrap my head around is the really complicated presentations that affine Lie algebras have and how one is supposed to do anything with them.”

    I’ve been wondering the same thing for pretty much anything related to quantum algebra.

  2. Great! I’m in the process of reading, but since I have to take a break anyway, here’s a minor confusion: when you define Fock space as ” the ring of symmetric functions over K in infinitely many variables” do you mean the direct sums of the rings of symmetric functions over k variables, as k–> infty? So the functions just have to be symmetric in some particular subset of the variables, right? Otherwise the set would be empty, maybe?

    Again I still haven’t finished the post, but does this affine lie algebra have anything to do with the temperley-lieb algebra (if you’ve even heard of it, it’s pretty obscure and unimportant I guess)? it’s suggestive because when you bracket t^1 and t^-1 (a trivial loop, so to speak) you kill it and stick in a central element to record the killing, just like in TL. I know at least it has to do with conformal field theory or something? Anyway I wish I knew this field better! And how loop groups fit into everything.


  3. Hi Sam,

    There’s two ways to think of the ring of symmetric functions in infinitely many variables. The first is to define it as the ring of formal power series in infinitely many variables of bounded degree which are invariant under arbitrary permutation of the variables (doesn’t matter if they are infinite permutations or finite support since every term is of finite degree). The other way is as follows: there is a map from the ring of symmetric functions in k variables to the ring of symmetric functions in k-1 variables obtained by setting the last variable equal to 0. Then we take the inverse limit of these maps in the category of graded rings (working in this category is crucial, or else we would get a much bigger ring).

    And I have no idea what the relation between the affine Lie algebra \widehat{\mathfrak{sl}_n} and the Temperley–Lieb algebra is, sorry.

  4. The Temperley-Lieb algebra has little to do with the affine part and everything to do with the q part. Namely, it is the centralizer algebra of the U_q(sl_2) (no affine) action on the space of rank n tensors (\mathbb{C}^2)^{\otimes n}.

    Also I was hoping to read more about this in Leclerc’s paper but unfortunately the link is broken.

  5. Jon: I’ve uploaded the paper to my website and changed the link.

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