On gamma-vectors and Chow polynomials of restrictions of reflection arrangements

TL;DR

This work investigates two polynomial invariants attached to restrictions of reflection arrangements: the $h$-polynomial with its $\gamma$-vector and the Chow polynomial. It proves that all restrictions of reflection arrangements are $\gamma$-positive and provides an explicit combinatorial formula for the Chow polynomial in type $B$, while showing that the intermediate $D_{n,s}$ restrictions interpolate arithmetically between types $B$ and $D$. The authors deploy two complementary methods to establish Chow polynomial arithmeticity for $\mathcal{D}_{n,s}$: (i) an EL-labeling-based approach yielding a linear interpolation $H_{\mathcal{D}_{n,s}}(t)=\frac{s}{n}H_{\mathcal{B}_n}(t)+\frac{n-s}{n}H_{\mathcal{D}_n}(t)$, and (ii) a characteristic-polynomial/minor-based recursive approach that recovers the same interpolation. Additionally, the work extends γ-positivity to all restrictions, including computational verification for exceptional cases, and supplies explicit $\gamma$-vectors and Chow polynomials up to moderate ranks, connecting type $B$ and type $D$ phenomena within a unified combinatorial framework.

Abstract

Simplicial arrangements are a special class of hyperplane arrangements, having the property that every chamber is a simplicial cone. It is known that the simpliciality property is preserved under taking restrictions. In this article we focus on the class of reflection arrangements and investigate two different polynomial invariants associated to them and their restrictions, the $h$-polynomial with its $γ$-vector and the Chow polynomial. We prove that all restrictions of reflection arrangements are $γ$-positive and give an explicit combinatorial formula of the Chow polynomial in type $B$. Furthermore we prove that for a special class of restrictions of arrangements of type $D$, called intermediate arrangements, both the $h$-polynomial as well as the Chow polynomial behave arithmetically, that is they interpolate linearly between the respective invariants for type $B$ and $D$.

Figures (1)

• Figure 1: $\mathcal{L}(\mathcal{D}_3)$ with the EL-labeling.

Theorems & Definitions (118)

• Conjecture 1.1
• Theorem A
• Theorem B
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7