Gale and Koszul duality, together at last July 14, 2008Posted by Ben Webster in category O, combinatorics, hyperplanes, mathematical physics, papers, the arXiv.
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 (or equivalently, an algebra) from a linear program (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. 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 is , the category associated to the Gale dual of .
Now, part of the data of a linear program is an “objective function” (which we’ll denote by ) and of bounds for the contraints (which will be encoded by a vector ). 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 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 , which we conjecture to include the fundamental group of the space of generic choices of objective function.
Hypertoric varieties and Koszul duality April 4, 2008Posted by Ben Webster in Algebraic Geometry, category O, combinatorics, D-modules, homological algebra, IAS, representation theory, talks.
So, on Wednesday, I gave a talk with the above title at IAS, about work in progress with Tom Braden, Tony Licata, and Nick Proudfoot. I was hoping to get David Nadler to blog it for me, but he was *ahem* indisposed. Failing that, I’ll direct you all to David Ben-Zvi’s notes (warning: freaking huge PDF). Hopefully, that will whet your appetite for the forthcoming paper.
Interpreting the Hecke Algebra February 25, 2008Posted by David Speyer in Algebraic Geometry, category O, combinatorics.
Let denote the finite field with elements. Let denote the set of complete flags in . Such a flag is a chain of subspaces of , with and . For example, if , an element of is a quadruple , where is a plane through the origin and is a line through the origin, with contained in . Of course, including and is redundant, as is always zero and is always , but we will see that it is convenient to have them around. Now consider the following operation: forget and replace it by one of the other possible -planes lying between and . There are such choices, and we pick uniformly at random. Note that we are forbidden to just put back where it was. We will denote this operation by .
Let’s do an example: starting with a given flag, in the case , what is the effect of ? Note that we draw our pictures projectively, so we visualize as a pair (point, line through the point) in the projective plane. Let’s call this pair . The first step moves the point along the line to a new point . The second step pivots the line about the point to a new line . And the final step slides the point along the pivoted line to its final resting place at .
(Click the image for a larger version.) Now, if we are given and , what is the probability that takes us from to ?
Exercise: This probability is zero if , or if any two of the points , and coincide. If none of these unfortunate events occur, then the probability is . (Hint: note that we must take to have any chance of success.
Now, here is the interesting thing. Run through the same analysis for . 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 the rest of the time.
So we can write the equation and, more generally, . If you are uncomfortable talking about an equality of “processes”, you may interpret as a linear operator on the space of probability distributions on . There are also two other relations between the . The first, which is completely obvious, is that when . The second, which takes only a moment’s more thought, says that . In other words, if we do twice, there is a chance that we will get back the -plane we stared with and a 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.
Dispatches from the Conference on Gauge and Representation Theory November 27, 2007Posted by Ben Webster in category O, conferences, D-modules, geometric Langlands, talks.
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…. (more…)
Koszul algebras and Koszul duality November 1, 2007Posted by Ben Webster in category O, homological algebra.
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 is an algebraically closed field, any semi-simple -algebra is a product of matrix algebras over .
(We say that an algebra is semi-simple if any submodule of any -module has a complement, that is, if every short exact sequence of -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. (more…)
John Brundan and the centers of blocks of category O June 27, 2007Posted by Ben Webster in Algebraic Topology, category O, Category Theory.
So, those of you who like the categorical approach to representation theory probably know about the center of a category:
Definition. The center of an abelian category is the rings of endomorphisms of the identity functor. That is, an element of the center is a system of maps for all objects in your category such that for ANY morphism , we have .