Date:
Tue, 20/06/202312:00-13:00
Location:
Levy 7 Hall and Zoom
Zoom Link:
https://huji.zoom.us/j/84511564169?pwd=SkJmY285YnFIWkZqNmxuaVZsVVQ2UT09
Meeting ID: 845 1156 4169
Passcode: 171220
Title: On the enumeration of Shi regions in Weyl cones
Abstract: Let Φ be an irreducible crystallographic root system with Weyl group W spanning a Euclidean space V . The reflection arrangement of W , which is the collection of hyperplanes ⟨a, x⟩ = 0 with a ∈ Φ, partitions the spaceV into cones known as Weyl cones. If Φ^+ is the positive part of Φ, the Shi arrangement is the collection of hyperplanes ⟨a, x⟩ = 0, 1 with a ∈ Φ^+ , which partition V into Shi regions. Since the Shi arrangement contains the reflection arrangement, each Weyl cone is partitioned by Shi regions. Our main goal is to determine the number of Shi regions in each Weyl cone. To do so, we explain how Shi regions within each Weyl cone biject to antichains of a naturally-defined subposet of the root poset (Φ^+ , ⪯). Then, we associate the root poset (Φ^+ , ⪯) to an acyclic directed graph Γ_Φ so that antichains in each of the above subposets are in bijection with paths in Γ_Φ avoiding a certain collection of subpaths. We conclude with a determinental formula which resolves our path counting. This is joint work with Aram Dermenjian.
https://huji.zoom.us/j/84511564169?pwd=SkJmY285YnFIWkZqNmxuaVZsVVQ2UT09
Meeting ID: 845 1156 4169
Passcode: 171220
Title: On the enumeration of Shi regions in Weyl cones
Abstract: Let Φ be an irreducible crystallographic root system with Weyl group W spanning a Euclidean space V . The reflection arrangement of W , which is the collection of hyperplanes ⟨a, x⟩ = 0 with a ∈ Φ, partitions the spaceV into cones known as Weyl cones. If Φ^+ is the positive part of Φ, the Shi arrangement is the collection of hyperplanes ⟨a, x⟩ = 0, 1 with a ∈ Φ^+ , which partition V into Shi regions. Since the Shi arrangement contains the reflection arrangement, each Weyl cone is partitioned by Shi regions. Our main goal is to determine the number of Shi regions in each Weyl cone. To do so, we explain how Shi regions within each Weyl cone biject to antichains of a naturally-defined subposet of the root poset (Φ^+ , ⪯). Then, we associate the root poset (Φ^+ , ⪯) to an acyclic directed graph Γ_Φ so that antichains in each of the above subposets are in bijection with paths in Γ_Φ avoiding a certain collection of subpaths. We conclude with a determinental formula which resolves our path counting. This is joint work with Aram Dermenjian.