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

TL;DR

This paper computes characteristic quasi-polynomials for Coxeter arrangements of types A, B, C, and D and their deletions, revealing factorization properties and the inductive structure of related hypertori posets.

Contribution

It provides explicit calculations of characteristic quasi-polynomials for specific Coxeter arrangements and their deletions, demonstrating their factorization and inductive poset structure.

Findings

Characteristic quasi-polynomials are computed for types A, B, C, D arrangements.

The quasi-polynomials for deletion arrangements are shown to factorize.

The hypertori poset of the toric arrangement is proven to be inductive.

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.