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

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.

💡 Research Summary

The paper investigates characteristic quasi‑polynomials of Shi arrangements of types C and D, focusing on the effect of fixing or deleting a single hyperplane. A characteristic quasi‑polynomial counts the number of points in the complement of a hyperplane arrangement over the finite ring ℤ/qℤ; it is known to be a quasi‑polynomial in q with a period equal to the least common multiple (lcm) of certain integer invariants of the arrangement. Earlier work by the first author and collaborators computed these quasi‑polynomials for the type B Shi arrangement Bₘ, obtaining |M(Bₘ,q)| = (q − 2m)ᵐ and showing a period collapse (the minimal period is 1 while the lcm‑period is 2). The present work extends this analysis to the type C and type D Shi arrangements, which are not simply related to the type B case because the defining integer matrices have determinant 2, so a change of basis from the simple‑root basis to the standard basis is not unimodular.

Type C.
The authors adapt Athanasiadis’s counting method, constructing a visual “boxes‑and‑circles” representation that distinguishes the cases where q is odd or even. For odd q, the condition 2xᵢ = 1 forces xᵢ = (q+1)/2, which leads to a specific forbidden box in the diagram; for even q the condition disappears. By carefully arranging the numbers 1,…,m in the boxes and counting admissible configurations, they prove that |M(Cₘ,q)| = (q − 2m)ᵐ for all sufficiently large q, matching the type B result but derived independently. They then compute the characteristic quasi‑polynomials of every restriction C_Hₘ obtained by fixing a hyperplane H. The formulas involve the auxiliary variable T = q − 2m and differ according to the parity of q. For example, for H = {2xᵢ = 0} one gets (T m−i)(T+1)^{i−1} when q is odd and 2·(T m−i)(T+1)^{i−1} when q is even; for H = {x_i−x_j = 0} the expression is (T+1)^{j−i−1}(T+2)^{m−j+i}, independent of parity. All these results are collected in Theorem 1.6.

Type D.
For the type D arrangement Dₘ, the authors exploit a direct bijection with the type B arrangement shifted by two units in q. Specifically, they construct a map between M(D_Hₘ,q) and M(B_{H′}ₘ,q+2) that respects the hyperplane equations, allowing them to import the known type B quasi‑polynomials. Consequently, the quasi‑polynomials for D_Hₘ involve the shifted variables (T+3), (T+4), etc., where again T = q − 2m. Theorem 1.7 lists the explicit formulas for the four families of hyperplanes (x_i−x_j = 0, x_i−x_j = 1, x_i+x_j = 0, x_i+x_j = 1), each with separate expressions for odd and even q.

Period collapse and deletions.
Because the quasi‑polynomials for both types C and D reduce to ordinary polynomials (minimal period 1) while the lcm‑period is known to be 2, a complete period collapse occurs. Using the deletion–restriction relation |M(A_q)| = |M(A_q{H_q})| − |M(A_H_q)|, the authors translate the restriction results into statements about deletions. They obtain a full classification of hyperplanes whose deletion yields a genuine polynomial characteristic quasi‑polynomial: for type C only hyperplanes of the form x_i−x_j = 0 or x_i−x_j = 1; for type D a more intricate list involving symmetric pairs (x_i−x_{m+1−i}=0), the “1‑shift” hyperplanes x_i−x_j = 1, and certain sum‑type hyperplanes x_1+x_j = 0 and x_i+x_m = 1. Moreover, they analyze the case of deleting two parallel hyperplanes and give necessary and sufficient conditions (Corollary 1.9).

Implications.
The work demonstrates that the characteristic quasi‑polynomials of Shi arrangements are highly sensitive to the underlying root system and the choice of basis, yet they exhibit a uniform pattern of period collapse across types B, C, and D. The bijective reduction from type D to type B suggests a broader strategy for handling other Coxeter types (e.g., E, F, G) where similar determinant‑2 transformations appear. The explicit formulas also provide concrete data for testing conjectures about the relationship between characteristic polynomials (the first constituent of the quasi‑polynomial) and the full quasi‑polynomial, and they may be useful in enumerative problems involving lattice points in polyhedral complements, toric arrangements, and related combinatorial geometry contexts.

In summary, the paper delivers a complete, explicit description of characteristic quasi‑polynomials for deletions of Shi arrangements of types C and D, establishes the occurrence of period collapse, and clarifies precisely which deletions preserve polynomiality, thereby extending the combinatorial theory of Shi arrangements and opening avenues for further exploration in Coxeter‑type hyperplane arrangement theory.