A hunka hunka burnin’ knot homology

One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?

This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.

At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). Continue reading


Symplectic duality slides

I’ve been too lazy to write in detail about the progress in my research (well, I am writing six papers and applying to jobs, so it isn’t entirely due to laziness), but I did recently speak in the symplectic seminar at MIT, and have posted the slides on my webpage. Obviously, they’re less useful without someone to explain them, but given the current lack of an overarching paper on the subject (that’s no. 5 on the list, I promise), I thought it might be edifying. Executive summary below the cut. Continue reading

Gale and Koszul duality, together at last

So, in past posts, I’ve attempted to explain a bit about Gale duality and about Koszul duality, so now I feel like I should try to explain what they have to do with each other, since I (and some other people) just posted a preprint called “Gale duality and Koszul duality” to the arXiv.

The short version is this: we describe a way of getting a category \mathcal{C}(\mathcal{V}) (or equivalently, an algebra) from a linear program \mathcal{V} (or as we call it, a polarized hyperplane arrangement).

Before describing the construction of this category, let me tell you some of the properties that make it appealing.

Theorem. \mathcal{C}(\mathcal{V}) is Koszul (that is, it can be given a grading for which the induced grading on the Ext-algebra of the simples matches the homological grading).

In fact, this category satisfies a somewhat stronger property: it is standard Koszul (as defined by Ágoston, Dlab and Lukács.  Those of you with Springer access can get the paper here).  In short, the category has a special set of objects called “standard modules” (which you should think of as analogous to Verma modules) which make it a “highest weight category,”  such that these modules are sent by Koszul duality to a set of standards for the Koszul dual.

Of course, whenever confronted with a Koszul category, we immediately ask ourselves what its Koszul dual is.  In our case, there is a rather nice answer.

Theorem. The Koszul dual to \mathcal{C}(\mathcal{V}) is \mathcal{C}(\mathcal{V}^\vee), the category associated to the Gale dual \mathcal{V}^\vee of \mathcal{V}.

Now, part of the data of a linear program is an “objective function” (which we’ll denote by \xi) and of bounds for the contraints (which will be encoded by a vector \eta).  Stripping these way, we end up with a vector arrangement, simply a choice of a set of vectors in a vector space, which will specify the constraints.

Theorem. If two linear programs have same underlying vector arrangment, the categories \mathcal C(\mathcal V) may not be equivalent, but they will be derived equivalent, that is, their bounded derived categories will be equivalent.

Interestingly, these equivalences are far from being canonical. In the course of their construction, one actually obtains a large group of auto-equivalences acting on the derived category of \mathcal{C}(\mathcal{V}), which we conjecture to include the fundamental group of the space of generic choices of objective function.

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Interpreting the Hecke Algebra

Let F_q denote the finite field with q elements. Let Flag_n(q) denote the set of complete flags (V_0, V_1, \ldots, V_n) in F_q^n. Such a flag is a chain of subspaces of F_g^n, with V_{i} subset V_{i+1} and dim V_i=i. For example, if n=3, an element of Flag_n(q) is a quadruple ( { 0 }, L, H, F_q^3), where H is a plane through the origin and L is a line through the origin, with L contained in H. Of course, including V_0 and V_n is redundant, as V_0 is always zero and V_n is always F_q^n, but we will see that it is convenient to have them around. Now consider the following operation: forget V_i and replace it by one of the other possible i-planes lying between V_{i-1} and V_{i+1}. There are q such choices, and we pick uniformly at random. Note that we are forbidden to just put V_i back where it was. We will denote this operation by P_i.

Let’s do an example: starting with a given flag, in the case n=3, what is the effect of P_1 P_2 P_1? Note that we draw our pictures projectively, so we visualize (V_1, V_2) as a pair (point, line through the point) in the projective plane. Let’s call this pair (p,\ell). The first step moves the point p along the line \ell to a new point p' . The second step pivots the line \ell about the point p' to a new line \ell'. And the final step slides the point p' along the pivoted line \ell' to its final resting place at p''.


(Click the image for a larger version.) Now, if we are given (p, \ell) and (p'', \ell'), what is the probability that P_1 P_2 P_1 takes us from (p, \ell) to (p'', \ell')?

Exercise: This probability is zero if \ell=\ell', or if any two of the points p, \ell \cap \ell' and p'' coincide. If none of these unfortunate events occur, then the probability is 1/q^3. (Hint: note that we must take p'=\ell \cap \ell' to have any chance of success.

Now, here is the interesting thing. Run through the same analysis for P_2 P_1 P_2. The intermediate stages of the process will look completely different, but the end effect is precisely the same: probability zero in the degenerate cases mentioned above, and probability 1/q^3 the rest of the time.


So we can write the equation P_1 P_2 P_1=P_2 P_1 P_2 and, more generally, P_{i} P_{i+1} P_{i}=P_{i+1} P_i P_{i+1}. If you are uncomfortable talking about an equality of “processes”, you may interpret P_i as a linear operator on the space of probability distributions on Flag_n(q). There are also two other relations between the P_i. The first, which is completely obvious, is that P_i P_j=P_j P_i when |i-j| geq 2. The second, which takes only a moment’s more thought, says that P_i^2=1/q+(1-1/q) P_i. In other words, if we do P_i twice, there is a 1/q chance that we will get back the i-plane we stared with and a 1-1/q chance that we won’t. Noah Snyder and John Baez can talk for hours about how this shows up in the tensor category of representations of a quantum group, so I won’t. Instead, I’ll explain some of the consequences of this for combinatorialists. At the end, I’ll bring up a very natural question that Jim Propp asked me, to which I don’t have a good answer yet. If no one writes in with a good solution, then I’ll pull together a sequel explaining my failed attempts.

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Dispatches from the Conference on Gauge and Representation Theory

So, I got back from Scotland and my family Thanksgiving (mmm, turkey) just in time for an enormous conference here in Princeton on Gauge and Representation Theory. I don’t really have the energy at the moment for some full blown conference-blogging, but I thought I would mention what was going on for the benefit of those who couldn’t attend.

(EDIT: For those of you who find me too vague, David Ben-Zvi has posted some notes from the talks here.)

I’m afraid I can’t really comment on the physics talks. While my level of total lostness varied during them, the average was high enough that I have nothing intelligent to say. Peanut gallery?

On the subject of talks I did understand some of…. Continue reading

Koszul algebras and Koszul duality

One of the famous theorems that tend to crop up in undergraduate algebra classes is the Artin-Wedderburn theorem, which says

Theorem. Any semi-simple ring is a product of matrix algebras over division algebras. In particular, if k is an algebraically closed field, any semi-simple k-algebra is a product of matrix algebras over k.

(We say that an algebra A is semi-simple if any submodule of any A-module has a complement, that is, if every short exact sequence of A-modules splits).

Now, looking at this theorem, one might imagine that we now know a lot about finite-dimensional algebra. After all, there are only two kinds of finite-dimensional algebras, semi-simple and non-semi-simple, and we understand one of those halves quite well. Better yet semi-simplicity is an “open” condition. If we think about the set of associative products a finite dimensional vector space could have, the set of such products which are semi-simple is an open set in the Zariski topology, which those of us who like algebraic geometry know means it is pretty darn big, provided it is non-empty (which is it is, since every vector space has a semi-simple product as the sum of a bunch of copies of the field).

But, of course, this is ridiculous. To borrow a metaphor, dividing algebras into semi-simple and not-semi-simple is like dividing the world into bananas and non-bananas. Continue reading

Components of Springer fibers, category O, and Khovanov’s “functor valued invariant of tangles”

I’ll just note at the beginning, this post is a bit of an experiment. At this point, it is about a semi-finished research thought of mine (which I’m not 100% sure is original, but I’m putting it out on the internet at least in part hoping that the internet will be able to tell me whether it’s original or not), and will consequently probably be a bit more technical than the average post on this blog, but hopefully, at least a few of you will be able to follow me.

As many of you know, my co-blogger Joel recently posted a preprint (with Sabin Cautis), which constructs a knot homology theory using the geometry of coherent s heaves and Fourier-Mukai transforms on convolutions of minuscule orbits in the affine Grasmannian of SL_2.

On the other hand, last year, Catharina Stroppel published a couple of papers on the relationship between Khovanov’s original construction of “a functor valued invariant of tangles” and various flavors of category O. From what I understand, underlying this is a philosophy that the \mathfrak{sl}_n version of Khovanov-Rozansky will be related a block to category O that lies on a dimension n-1-dimensional wall of the Weyl chamber of \mathfrak {sl}_d (where d is a number relating to the number of strands in your tangle diagram).

One natural question leaps to mind: how are these related? Continue reading

Soergel bimodules

The next ingredient in my paper with Geordie is understanding a bit about Soergel bimodules.

Soergel bimodules (or “Soergelsche Bimoduln” auf Deutsch) are a remarkable category of bimodules over a polynomial ring. The main thing that’s remarkable about them is that they categorify the Hecke algebra of S_n (for those of you who don’t know any other Hecke algebras, pretend I just said “Hecke algebra”).

Theorem (Soergel). The split Grothendieck group of the category of Soergel bimodules for S_n is the Hecke algebra \mathcal{H}_n of S_n.

We’ll unpack that a bit later. For the moment, bear with me. Of course, one of the most important things about Grothendieck groups is that they have a natural basis. If one uses a split Grothendieck group (like a normal Grothendieck group, except we only have a relation that says [M]+[N]=[M\oplus N], not for non-trivial extensions), then this basis will be the classes of the indecomposable elements.

Theorem (Soergel). The basis of indecomposable objects is the famous basis of Kazhdan and Lusztig.

Now that I’ve told you why people like Soergel bimodules, I guess I had better define them.

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