Interactions between combinatorics, representation theory, and algebraic geometry

Geometric representation theory and orbifold Hilbert Schemes



This project hopes to explain some experimentally observed patterns in the combinatorics of partitions in terms of representations of certain algebras. This would both prove some conjectures, and lead the path forward to developing them further.


Imagine the possible configurations of n indistinguishable points in the plane. When all the points are distinct, this space looks like 2n dimensional space -- we have two dimensions we can wiggle each of the n points. But if two of the points collide, we can no longer tell which point we're wiggling away, and the space becomes badly behaved, or "singular" there. The Hilbert scheme of n points is a closely related space that fixes these singularities by "remembering how the points ran together". It is a complicated space, but Ellingsrud and Stromme proved that it's shape or "topology" can be described in terms of simpler, combinatorial objects known as the partitions of n -- the ways of writing n as a sum of smaller numbers. This connection has proven useful both in understanding the Hilbet scheme of points, and in proving theorems about partitions.


Goettsche extended this result to when our n points are wandering around an arbitrary two dimensional space, instead of just the plane, and Nakajima and Grojnowski gave another proof of Goettsche's result using ideas from geometric representation theory.


It turns out that similar patterns to those found by Ellingsrud and Stromme hold if we look at "Orbifold Hilbert schemes", where we require the n points to satisfy certain symmetries. Some of these patterns have been proven geometrically, but some remain conjectures, and though all of them look like they could have proofs coming from geometric representation theory. In fact, special cases of the Orbifold Hilbert schemes have already been studied extensively by Nakajima under a different name, and led to huge breakthroughs in representation theory. This project will make the first steps in extending the Geometric Representation Theory story beyond Nakajima's examples.


In particular, the student will prove an analog of Nakajima and Grojnowski's Heisenberg algebra result for Orbifold Hilbert schemes, building bridges between the separate areas of combinatorics, representation theory, and algebraic geometry.