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.

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