Characteristic quasi-polynomials of deletions of Shi arrangements of type C and type D
Akihiro Higashitani, Masato Konoike, Norihiro Nakashima, Satoshi Ono
TL;DR
The paper computes characteristic quasi-polynomials for deletions of Shi arrangements of types C and D. It adapts the counting framework from type B to type C, deriving parity-dependent restriction formulas and showing the overall quasi-polynomials collapse to polynomials for certain deletions. For type D, it establishes a concrete bijection with type B to transfer known results and obtain explicit restrictions across all hyperplanes, enabling a complete description of period collapse phenomena. Together, the results illuminate how deletions interact with lcm periods and reveal structural differences between the B, C, and D Shi arrangements at the level of restriction counts and intersection posets.
Abstract
Characteristic quasi-polynomials enumerate the number of points in the complement of hyperplane arrangements modulo positive integers. In this paper, we compute the characteristic quasi-polynomials of the restrictions of the Shi arrangements of type C and type D by one given hyperplane, respectively. The case of type C is established by extending the method developed in our previous work on type B (\cite{HN2024}), while the case of type D is deduced through a direct connection with the results on type B. As a corollary, we determine whether period collapse occurs in the characteristic quasi-polynomials of the deletions of the Shi arrangements of type C and type D.