Special Year Seminar I

We consider the space of configurations of n points in the three-sphere $S^3$, some of which may coincide and some of which may not, up to the free and transitive action of $SU(2)$ on $S^3$. 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.