Shi arrangements restricted to Weyl cones

Abstract

We consider the restrictions of Shi arrangements to Weyl cones, their relations to antichains in the root poset, and their intersection posets. For any Weyl cone, we provide bijections between regions, flats intersecting the cone, and antichains of a naturally-defined subposet of the root poset. This gives a refinement of the parking function numbers via the Poincar\'e polynomials of the intersection posets of all Weyl cones. Finally, we interpret these Poincar\'e polynomials as the Hilbert series of three isomorphic graded rings. One of these rings arises from the Varchenko-Gel'fand ring, another is the coordinate ring of the vertices of the order polytope of a subposet of the root poset, and the third is purely combinatorial.