Characteristic elements for real hyperplane arrangements

TL;DR

This paper introduces characteristic elements in the Tits algebra of real hyperplane arrangements, linking them to the characteristic polynomial and deriving key combinatorial and geometric results.

Contribution

It defines characteristic elements, explores their properties, and applies them to obtain classical and new results on the characteristic polynomial of arrangements.

Findings

Derived Zaslavsky's formulas using characteristic elements

Connected characteristic elements to intrinsic volumes of faces

Constructed explicit examples of characteristic elements

Abstract

Characteristic elements of the Tits algebra of a real hyperplane arrangement carry information about the characteristic polynomial. We present this notion and its basic properties, and apply it to derive various results about the characteristic polynomial of an arrangement, from Zaslavsky's formulas to more recent results of Kung and of Klivans and Swartz. We construct several examples of characteristic elements, including one in terms of intrinsic volumes of faces of the arrangement.