Shi arrangements restricted to Weyl cones
Galen Dorpalen-Barry, Christian Stump
TL;DR
The paper develops a detailed correspondence for Shi arrangements restricted to Weyl cones, establishing bijections between Weyl-cone Shi regions, flats intersecting the cone, and root-poset antichains within appropriate subposets. It shows that the Poincaré polynomials of these cones encode Whitney numbers and Narayana refinements, and that summing over Weyl elements recovers the classical $(h+1)^{\ell}$ count; it further provides an algebraic realization by proving the Varchenko–Gel'fand ring and an order-ring are isomorphic and interprets their Hilbert series as the Poincaré polynomials, tying combinatorics to the geometry of order polytopes. The results extend to Shi deletions and yield a robust framework for understanding the interplay among regions, intersection posets, and antichains, with concrete demonstrations in low rank examples. This unifies combinatorial, geometric, and algebraic perspectives on Weyl-cone restrictions of Shi arrangements and offers tools for refined counting via Whitney/Narayana data and ring-theoretic interpretations.
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 Poincaré polynomials of the intersection posets of all Weyl cones. Finally, we interpret these Poincaré 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.