The Tutte Polynomial of Symmetric Hyperplane Arrangements

TL;DR

This paper extends the computation of the Tutte polynomial to complex and symmetric hyperplane arrangements, enabling the analysis of arrangements linked to imprimitive reflection groups.

Contribution

It introduces a method to compute the Tutte polynomial for symmetric hyperplane arrangements, broadening the scope beyond real arrangements and classical Weyl groups.

Findings

Extended Tutte polynomial computation to complex arrangements.

Derived Tutte polynomials for arrangements related to imprimitive reflection groups.

Provided a framework for analyzing symmetric hyperplane arrangements.

Abstract

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This article has two objectives: On one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.