Enumerating regions of Shi arrangements per Weyl Cone

TL;DR

The paper develops a uniform, determinant-based method to count Shi-ar arrangement regions inside each Weyl cone $C_w$ for finite Weyl groups. By leveraging Shi’s antichain correspondence, Armstrong–Reiner–Rhoades’ subposet construction, and Shi diagrams, it builds acyclic digraphs whose $I\to F$ paths biject with antichains; counting paths avoiding inversions $N(w^{-1})$ yields a determinantal formula for $|C_w|$. The method is carried out explicitly for types $A$, $B$, and $D$ (with extensions to exceptional types $E_6$, $F_4$, $G_2$ and open questions for $E_7$, $E_8$), including a variant that encodes Narayana/Poincaré refinements by tracking corners. This approach unifies region enumeration across Weyl types, providing computationally efficient determinants thanks to sparse matrix structure. The work advances precise, cone-wise Shi-region counts and offers refined combinatorial statistics aligned with root-poset antichains and lattice-path enumerations.

Abstract

Given a Shi arrangement $\mathcal{A}_Φ$, it is well-known that the total number of regions is counted by the parking number of type $Φ$ and the total number of regions in the dominant cone is given by the Catalan number of type $Φ$. In the case of the latter, Shi gave a bijection between antichains in the root poset of $Φ$ and the regions in the dominant cone. This result was later extended by Armstrong, Reiner and Rhoades where they gave a bijection between the number of regions contained in an arbitrary Weyl cone $C_w$ in $\mathcal{A}_Φ$ and certain subposets of the root poset. In this article we expand on these results by giving a determinental formula for the precise number of regions in $C_w$ using paths in certain digraphs related to Shi diagrams.

Figures (5)

• Figure 1: The diagrams $\Lambda_{A_3}$ (left), $\Lambda_{B_3}$ (middle) and $\Lambda_{D_5}$ (right) as constructed by Shi.
• Figure 2: The digraph $\Gamma_{A_7}$ (left) and $\Gamma_{B_4}$ (right) associated to the Shi arrangements of type $A$ and $B$.
• Figure 3: On the left is the subdiagram associated to the antichain $\left\{ \alpha_{23} \right\}$ and on the right is the subdiagram associated to the antichain $\left\{ \alpha_{13, 55} \right\}$.
• Figure 4: The antichains $\{\alpha_{24},\alpha_{55}\}$ and $\{\alpha_{23,55},\alpha_{44}\}$ in the root poset $D_5$ are represented by subdiagrams which are shaded in the diagram $\Lambda_{D_5}$. In the graph $\Gamma_{D_5}$ they corresppond to distinct paths, each having exactly two corners.
• Figure 5: The diagram $\Gamma_{D_5}$ for the $D_5$ Shi arrangement.

Theorems & Definitions (35)

• Theorem 1.1
• Example 1
• Example 2
• Theorem 2.1: Shi_Number
• Theorem 2.2: ArmstrongReinerRhoades_Parking
• Example 3
• Theorem 3.1
• proof
• Theorem 3.2
• proof