Zonotopal Algebras, Configuration Spaces, and More

We consider the space of configurations of n points in the three-sphere S3, some of which may coincide and some of which may not, up to the free and transitive action of SU(2) on S3. We prove that the cohomology ring with rational coefficients is isomorphic to an internal zonotopal algebra, which is a combinatorially defined ring appearing independently in the work of Holtz and Ron and of Ardila and Postnikov. We use zonotopal algebras to prove a conjecture of Moseley, Proudfoot, and Young in 2016 about the cohomology of these configuration spaces. Along the way, we also give a formula for the equivariant K-polynomial of a matroid Schubert variety with respect to a finite symmetry group.

Based on joint work with Galen Dorpalen-Barry, André Henriques, and Nicholas Proudfoot.