Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characteristic quasi-polynomials for deformations of Coxeter arrangements of types A, B, C, and D

Yusuke Mori, Norihiro Nakashima

Abstract

Kamiya, Takemura, and Terao introduced a characteristic quasi-polynomial which enumerates the numbers of elements in the complement of hyperplane arrangements modulo positive integers. In this paper, we compute the characteristic quasi-polynomials for specific arrangements which contain the Coxeter arrangements of types A, B, C, and D described by the orthonormal basis. We also compute the characteristic quasi-polynomials for their deletion arrangements and we can show that they are factorized.From this result, the poset generated by hypertori of the corresponding toric arrangement is an inductive poset.

Characteristic quasi-polynomials for deformations of Coxeter arrangements of types A, B, C, and D

Abstract

Kamiya, Takemura, and Terao introduced a characteristic quasi-polynomial which enumerates the numbers of elements in the complement of hyperplane arrangements modulo positive integers. In this paper, we compute the characteristic quasi-polynomials for specific arrangements which contain the Coxeter arrangements of types A, B, C, and D described by the orthonormal basis. We also compute the characteristic quasi-polynomials for their deletion arrangements and we can show that they are factorized.From this result, the poset generated by hypertori of the corresponding toric arrangement is an inductive poset.
Paper Structure (10 sections, 18 theorems, 121 equations)

This paper contains 10 sections, 18 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characteristic quasi polynomials for deformation of Coxeter arrangements
  4. Deformation of type A
  5. Deformation of type D
  6. Deleting hyperplanes
  7. A sequence of subarrangements of deformation of type A
  8. A sequence of subarrangements of deformation of type D
  9. Inductive poset
  10. Other examples

Key Result

Theorem 2.1

The function $|M_{\mathscr{A}_q}(q)|$ is a monic quasi-polynomial with a period $\rho_{\mathscr{A}}$ having the gcd property with respect to $\rho_{\mathscr{A}}$.

Theorems & Definitions (38)

  • Theorem 2.1: Kamiya, Takemura, and Terao KTT08KTT11
  • Theorem 2.2: Higashitani, Tran, and Yoshinaga HTY2112
  • Theorem 2.3: Kamiya, Takemura, and Terao KTT10-arxiv-v1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 3.3
  • Lemma 3.4
  • proof
  • ...and 28 more