Shi arrangements restricted to Weyl cones

TL;DR

This paper explores the structure of Shi arrangements within Weyl cones, establishing bijections with antichains, refining parking function counts, and connecting these to algebraic and combinatorial invariants.

Contribution

It introduces a detailed analysis of Shi arrangements restricted to Weyl cones, linking regions, flats, and antichains, and relates their intersection posets to graded rings and combinatorial structures.

Findings

Bijections between regions, flats, and antichains in Weyl cones.

Refinement of parking function enumeration via Poincaré polynomials.

Identification of three isomorphic graded rings related to the arrangements.

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.