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Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan

Takuro Abe, Tan Nhat Tran

Abstract

Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this paper, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This is a completion of the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems.

Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan

Abstract

Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this paper, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This is a completion of the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems.
Paper Structure (9 sections, 19 theorems, 46 equations, 3 figures)

This paper contains 9 sections, 19 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Back ground and motivation
  3. The main results
  4. Preliminaries
  5. Free arrangements
  6. Characteristic quasi-polynomials and Worpitzky-compatibility
  7. Proof of the first main result: Theorem \ref{['thm:WC']}
  8. Proof of the second main result: Theorem \ref{['thm:SF-characterize']}
  9. Further comments and remarks

Key Result

Theorem 1.2

Yo10 Let $\Phi$ be an irreducible root system of type $ADE$ and $\Sigma\subseteq\Phi^+$. Then $\Sigma$ is Shi-free if and only if $\Sigma$ is free and coclosed.

Figures (3)

  • Figure 1: Root system of type $A_2$ from Example \ref{['ex:A2']}.
  • Figure 2: The Worpitzky partition of the fundamental parallelepiped $P^\diamondsuit$ in type $G_2$. A non-facet intersection between an affine hyperplane and an upper closed alcove occurs only at the alcoves in green.
  • Figure 3: A pictorial illustration of the proof of Claim \ref{['cl:shift']}. Each root $\alpha$ in the root poset is written next to the evaluation $r(\alpha)$ of the map $r$. The illustration for Subcase 1 is on the left, the illustration for Subcase 2 is the entire figure.

Theorems & Definitions (38)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • Theorem 2.2
  • ...and 28 more