I want to write a post about the set-valued tableaux of Buch and how they are related to Schur polynomials. But since these two things are related to the K-theory and Chow ring, respectively, of the Grassmannian, I thought I would write a post explaining some basic generalities between the Chow ring and K-theory. Let X be a variety over an algebraically closed field K.

First let’s define the Chow groups. We first form the k-cycles to be the free Abelian group spanned by the k-dimensional subvarieties of X. Let [V] be the basis element corresponding to a subvariety V. Pick a subvariety W of X of dimension k+1, and a nonzero rational function f/g defined on W. If V is a codimension 1 subvariety of W, let be the ring obtained by taking the ring of polynomial functions on W and inverting all polynomial functions which are not identically zero on V. We define the order to be , where the dimension is as K-vector spaces. The divisor of f/g is given by . We say these divisors are rationally equivalent to 0, and define the Chow group to be the group of k-cycles modulo rational equivalence.

Let K be a field of characteristic 0, and let V be a vector space over K of dimension n, and pick k < n. Let X be the Grassmannian Grass(k, V). We’ll briefly explore the (bounded) derived category of coherent sheaves of X, denoted .

1. Derived category review

For those unfamiliar with derived categories, here’s a quick summary. If A is any Abelian category, set K(A) to be the category of (co)chain complexes of A with the morphisms being chain maps modulo homotopy equivalence. Chain maps which induce isomorphisms are formally inverted, and the result is the derived category of A. Usually we only want to consider bounded complexes, or at least complexes with finitely many nonzero (co)homology groups, and in this case we denote the category . The category is equipped with a shift functor, which just shifts the degrees of a given complex.

One thing we can do is reformulate derived functors. Given a left exact functor , we define its right derived functor as follows. Given an object X in A, an injective resolution of X becomes an isomorphism in (considering X as a complex with one nonzero term), so we define to be the complex obtained by applying F to . Actually, we don’t need an injective resolution, we only need a resolution consisting of F-acyclic objects (i.e., the usual right derived functors of F vanish for them). To define on a general complex , we need to find a double complex which is term by term an injective resolution for each (these are called Cartan–Eilenberg resolutions). Then we apply F to the total complex of . A similar story is true for right exact functors G, so we get left derived functors . For notation, the left derived functor of the tensor product is denoted .

In the last post of this series, I gave some definitions and facts regarding the Hall–Littlewood functions. I also sketched the relationship between symmetric functions and representations of the symmetric group. Now we’ll see how this works for the finite general linear groups.

We want to imitate the Frobenius character that is used to relate the characters of the symmetric group to the ring of symmetric functions. But since the description of the conjugacy classes of the finite general linear group (and hence the parametrization of its irreducible characters) are more complicated than the description for the symmetric group, we’ll need a bigger ring to work with.

I’ve been betraying the title of this blog and doing some abstract nonsense lately, mostly to relearn algebraic geometry for the -th time for some embarrassingly large . With the memory capabilities of a muffin, I am practically starting over each time. Luckily, each adventure feels slightly more beatable (exp points?), since I have a few more examples I can use for myself. Masnevets and I had a good discussion about a few basic examples of adjoint functors (recall the definition here: basically, we need a pair of functors and such that we have a natural isomorphism ), and thus we have a new Concrete Nonsense post.

Before we start, I want to state that I’m trying something new. This post is not intended to be an introduction to adjoints as I originally envisioned – I realized that there are many better sources for that. Instead, I’ll try to do a free association that juxtoposes a few elementary concepts. You don’t even have to know the definitions of adjoints to start seeing what I’m getting to, since I’ll be namedropping algebraic structures like Kanye West.

My first introduction to adjoints was from algebraic topology, where you naturally bump into the functors and . It was unnecessary at the time (for the scope of the course, at least) to know that they were adjoints, but now I know them as the “tensor-hom adjunction pair” (saying “tensor-hom” a lot helps me remember tensor as the left adjoint and hom as the right). Furthermore,  knowing this relationship allows me to remember some other things – in particular, knowing the left- and right- exactness of these functors, which I used to always mix up. Left adjoints are always right-exact, and right adjoints are always left-exact. Combined with knowing that tensoring is a left-adjoint, I now know that tensoring is right-exact and adjoints are left-exact.

Last time, I discussed how to parametrize the conjugacy classes of the general linear group G over a finite field. In a sense, this group is a q-analogue of the symmetric group, so we might try to imitate the constructions for the symmetric group to get information about G. There’s a nice construction that Frobenius worked out which connects the characters of the symmetric group with the combinatorics of the Schur functions. I’ll briefly summarize the statement. The conjugacy classes of the symmetric group on n letters are parametrized by partitions of n. So we can also parametrize the irreducible characters by partitions as well (though it is not clear how to do this in a “canonical” way a priori). Ignoring the indexing issue (which can be dealt with) and letting be the irreducible character indexed by evaluated at the conjugacy class consisting of permutations whose cycle lengths are given by the parts of , then one has where is a Schur function, is a power sum (Newton) symmetric function, and where is the number of times that i appears as a part of (the meaning of is that is the size of the conjugacy class index by .)

So the question to ask might be “can we find a similar interpretation for the characters of G?” The answer is yes, but becomes a bit more involved.

I’d like to start a series of posts, GLFq (general linear group over a finite field), in which I hope to explain the representation theory of (n and q will remain fixed). In this first post, I’ll explain how to write down the conjugacy classes of G. In later posts, I plan to introduce Hall–Littlewood polynomials and the characteristic map. I would like to also go into how to construct the actual representations, and discuss things related to Hall–Littlewood polynomials, like the q-Kostka polynomials and a lot of the interesting algebra/geometry behind them.

There are two pieces of data we would like to know. First, what is the size of G? Second, how do we parameterize the conjugacy classes? The first question is easy to answer since an invertible matrix is given by the data of n linearly independent vectors. The first one can be chosen to be any nonzero vector, so there are of them. In general, the ith one can be chosen to be any vector not in the span of the last i-1 (so we are just avoiding some i-1 dimensional subspace, which has elements), and hence there are choices for such a vector. All together, the number of elements of G is . We can rewrite this as to make it more analogous to the number of elements of the symmetric group.

Back to symplectic geometry. So far, everything I did in my last post only used the fact that the symplectic form was skew symmetric, not that it was closed. Indeed the “closed” property is rather mysterious, (as far as I’m concerned, although in the literature it is called “geometric”), since I don’t know of any really good geometric intuition for the action of exterior derivative on 2-forms. Still, it is a hugely important condition, and key to many of the special properties of symplectic geometry, notably for us, the Darboux theorem. Note that there is no real equivalent condition for Riemannian structures, and therefore it takes us in a whole new direction. I would love to have a better sense of how to “explain” why certain symplectic arguments don’t work in the Riemannian world (eg Darboux theorem), but I haven’t delved deeply enough into the proof to do this, since it’s not coherent to just say “the Riemannian form isn’t closed.”

One of my summer projects was to try to learn symplectic geometry. In this, my first installment of notes, I discuss some introductory notions; hopefully it’s not too rambling. In the continuation, I’ll prove Darboux’s theorem, a fundamental result which says that locally, all symplectic spaces are isomorphic (something which sharply distinguishes symplectic geometry from Riemannian geometry, where there are many local invariants, such as curvature).

EDIT: Here, I take the point of view that the reader is somewhat familiar with Riemannian geometry, and try to build intuition for the structures in symplectic geometry via an analogy with the Riemannian case. This was helpful for me, to some extent, in order to even have a chance of “breaking into” the field, so to speak. However, there are many reasons why this is possibly a misleading vantage point, so do not believe that it’s the whole story. It may be helpful for some. Please see the comments for additional (undoubtedly better, I am an extreme novice) points of view. Some of these would also be quite suitable for an introduction.

In both symplectic and Riemannian geometry, the main object of study is a smooth manifold equipped with a bilinear form on each tangent space, in such a way that the forms vary smoothly as we move between tangent spaces. In the (possibly more familiar) Riemannian case, this form is a symmetric, non-degenerate, positive definite form, turning each tangent space into a normed vector space. In symplectic geometry, we instead require a skew-symmetric bilinear form on each tangent space, again varying smoothly. We still require that at each point in our manifold , should be non-degenerate, so that if =0 for all , then must be 0. Finally, note that because is a skew-symmetric 2-form, it is a differential 2-form on , and we require that as a 2-form, is closed, i.e., . I’ll introduce examples as we go.

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.

Most of my “free math time” has been used to study for quals, but today I’ve made myself post to stop Steven from taking over this blog.

One of my favorite elementary algebric combinatorial results is the Matrix Tree Theorem, which states:

In a nondirected graph with vertices labelled , the number of spanning trees is equal to any principal minor of the Laplacian.

This cute result gets the number of trees on vertices () fairly quickly with some matrix manipulation, which I will leave as an exercise to the reader. I know two proofs of this theorem: the first one involves using the Cauchy-Binet formula on the Laplacian , after making the slick observation that , where is the incidence matrix. Another quick solution can be obtained by invoking the lesser-known version of the Matrix Tree Theorem for directed graphs, which is actually a bit simpler to prove:

In a directed graph with vertices labelled , the number of arborescences into vertex (that is, trees rooted at where all the edges point towards ) is equal to the -th minor of the Laplacian.

But this is not all!