Tutte Polynomials of Symmetric Hyperplane Arrangements

Abstract

Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal structure of a graph, and contains even other specializations from other sciences like the Jones polynomial in Knot theory, the partition function of the Pott model in statistical physics, and the reliability polynomial in network theory. In this article, we study the Tutte polynomial associated to more general objects which are the arrangements of hyperplanes. Indeed determining the Tutte polynomial of a graph is equivalent to determining the Tutte polynomial of a special hyperplane arrangement called graphic arrangement. In 2007 Ardila computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups, and the characteristic polynomials of the Catalan arrangements. The charateristic polynomial is also a specialization of the Tutte polynomial. In 2012 Seo computed the characteristic polynomials of the Shi threshold arrangements, and in 2017 Song computed the characteristic polynomials of the $\mathcal{I}_n$ arrangements in the plane and in the space. We aim to bring a more general result by introducing a wider class of hyperplane arrangements which is the set of symmetric hyperplane arrangements. We compute the Tutte polynomial of a symmetric hyperplane arrangement, and, as examples of application, we deduce the Tutte polynomials of the Catalan, the Shi threshold, and the $\mathcal{I}_n$ arrangements.