Characteristic quasi-polynomials of deletions of Shi arrangements of type C and type D

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.