The Tutte Polynomial of Ideal Arrangements

Abstract

The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.