Tutte Polynomial of Symmetric Hyperplane Arrangement II: Complex Hyperplane Arrangements

Hery Randriamaro

Published 2017-08-29Version 1

The Tutte polynomial is originally 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, like its number of forests and number of spanning subgraphs, 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 2007, Ardila defined the Tutte polynomial on more general objects which are the real hyperplane arrangements. Computing the Tutte polynomial of a graph is indeed equivalent to computing the Tutte polynomial of its graphic hyperplane arrangement. He computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups at the same time. De Concini and Procesi completed his results by computing the Tutte polynomials of the hyperplane arrangements associated to the exceptional Weyl groups one year later. At the beginning of 2017, Randriamaro introduced a wider class of hyperplane arrangements in the Euclidean spaces called symmetric hyperplane arrangements. He computed the Tutte polynomial of a symmetric hyperplane arrangement, which permits to deduce the Tutte polynomials of particular hyperplane arrangements like the Catalan, the Shi threshold, and the $\mathcal{I}_n$ arrangements. In this article, we propose to extend the investigation to the complex hyperplane arrangements. We compute the Tutte polynomial of a symmetric hyperplane arrangement in a Hermitian space. We particularly dedicate a section to the special subclass of colored symmetric hyperplane arrangements. From their Tutte polynomials, we deduce the Tutte polynomials of the hyperplane arrangements associated to the imprimitive reflection groups.