Enumerating Weyl Cones of Shi arrangements

We will give all background knowledge necessary to understand the talk. No prior experience with Coxeter groups or hyperplane arrangements is necessary.

The enumeration of regions in a Shi arrangement have a rich history starting from Shi in 1986 which showed that the Shi arrangements of type $A_n$ have $(n+1)^{(n-1)}$ regions. In order to refine this number, the Shi arrangement was split into Weyl cones where each cone is associated with an element in the associated Coxeter group. The dominant cone is the cone associated with the identity element and is counted by the Catalan numbers. By a result of Armstrong, Reiner and Rhoades, the other Weyl cones can be counted by the number of elements in the intersection of hyperplanes which intersect the cone. We continue this study by giving a formula to calculate the exact number of elements in each Weyl cone using a determinantal formula.

Bio
https://dermenjian.com/