Characteristic quasi-polynomials of deletions of Shi arrangements of type B and their period collapse

TL;DR

This paper computes characteristic quasi-polynomials for deletions of type B Shi arrangements, identifies when period collapse occurs, and resolves a related conjecture, advancing understanding of hyperplane arrangement enumerations.

Contribution

It provides explicit calculations of quasi-polynomials for these arrangements and fully characterizes period collapse, solving an open conjecture in the field.

Findings

Identified conditions for period collapse in type B Shi arrangements.

Explicit formulas for characteristic quasi-polynomials of deletions.

Confirmed the conjecture by Higashitani, Tran, and Yoshinaga.

Abstract

Characteristic quasi-polynomials are the enumerative functions counting the number of elements in the complement of hyperplane arrangements modulo positive integers. A notable phenomenon in this context is period collapse, where the quasi-polynomial reduces to a polynomial or has a smaller period than the lcm period. In this paper, we compute the characteristic quasi-polynomials of the restriction of the Shi arrangement of type B by one given hyperplane. As a corollary, we completely determine whether period collapse occurs in the characteristic quasi-polynomial of the deletion of the Shi arrangement of type B. This implies the solution for the conjecture posed by Higashitani, Tran and Yoshinaga in this case.