Lately I’ve been reading about quantum groups. I particularly enjoyed Bernard Leclerc’s paper Symmetric functions and the Fock space representation of , 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 , which is defined in the paper as the Lie algebra with generators () 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 of a simple Lie algebra as: with a Lie bracket given by , and then to say that is its universal central extension. More precisely, we say add a central element c, and then extend the bracket above via where is the Kronecker delta, and (,) is the Killing form of .
In the case that is , I want to discuss a more concrete (combinatorial) description. Just as can be thought as the traceless operators on an n-dimensional vector space (the standard representation), we can also find a standard representation for (the Fock space representation). For this, let denote the ring of symmetric functions over K in infinitely many variables. The Schur functions form a basis indexed by partitions, and this will be our representation. In order to describe the actions of on Sym, we’ll need some notation.
First, we represent partitions by their Young diagram ( 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 (resp. ) where the sum is over all obtained from by removing (resp. adding) an i-node, and define where is the number of 0-nodes of . Finally, set . Then is the Lie algebra spanned by these generators.
Unlike the case, the Fock space representation is not irreducible. Let be the power sum symmetric function where . It turns out that the set where are the highest weight vectors of this representation (i.e., they are killed by the , and are eigenvectors for d and the ). Furthermore, one has , so we have a natural notion of degree for our highest weight vectors. Within these graded subsets, the give an obvious choice of basis, but there is no reason to favor them: for example, also forms a basis for the highest weight vectors of degree 2. The point is that the Schur functions give a “natural basis” for 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 by its q-analogue , and similarly for . More precisely, we can’t deform the Lie algebra, but we can deform its universal enveloping algebra. The quantum group has generators , and and even more relations than has, so rather than give those, I just want to mention how to change the action on . Let q be a transcendental element over K, and let K(q) be the function field over K. We set , and to get the actions of the and on , 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 and be two partitions such that is obtained from by adding an i-node to it. Let (resp. ) be the number of indent i-nodes of (resp. number of removable i-nodes of ) which are strictly to the right of . Also set . Define the same numbers with the superscript r replaced by l by replacing “right” with “left.” Then we set (resp. ) where the sum is over all such that is an i-node (resp. is an i-node). We also set and define where is the number of removable and indent i-nodes of . And we can take the quantum group to be the K(q) algebra spanned by these generators.
We’ll use a family of operators 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 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 where the sum is over all such that 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”, reduces to multiplication by the plethysm .
We introduce a K-linear bar involution on via , and then extend this to a compatible K-linear involution on by requiring that it commute with the actions of and on , and that it fixes the basis vector . Let L (resp. ) be the free -submodule (resp. -submodule) of spanned by the basis . Then we have the following theorem.
Theorem. There exist two unique bar-invariant bases and of such that and .
The two bases are called the canonical basis and dual canonical basis of . They have a lot of nice properties. Going back to highest weight vectors, it turns out that is a highest weight vector for all . Furthermore, “setting ” this basis of highest weight vectors reduces to the plethysms (this is related to the classical limit of the operators ). Since it comes from a more “rigid” basis, we might be satisfied with this choice for a basis of highest weight vectors in . Another nice property which happens with canonical bases is a nonnegativity property: write and where the d and e are polynomials.
Theorem.The polynomials d and e have nonnegative coefficients as polynomials in q. Furthermore, is nonzero only if and similarly, is nonzero only if .
Here we are using the dominance order on partitions: if can be written as a nonnegative linear combination of vectors where 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.