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

TL;DR

This paper studies polynomial invariants of reflection arrangements, proving gamma-positivity for all restrictions and providing explicit formulas, especially in types B and D.

Contribution

It establishes gamma-positivity for restrictions of reflection arrangements and derives explicit formulas for Chow polynomials in type B.

Findings

All restrictions of reflection arrangements are gamma-positive.

Explicit combinatorial formula for Chow polynomial in type B.

In type D intermediate arrangements, invariants interpolate linearly between types B and D.

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 $\gamma$-vector and the Chow polynomial. We prove that all restrictions of reflection arrangements are $\gamma$-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 $D$.