The Tutte Polynomial of Ideal Arrangements
Hery Randriamaro (Madagaskar / Universität Kassel)
The Tutte polynomial was at the origin in 1954 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. Ardila extended in 2007 the notion of Tutte polynomial to hyperplane arrangements. At the same time, he computed the Tutte polynomial of the hyperplane arrangements associated with the root systems of the classical Weyl groups. The Tutte polynomials associated to the root systems of the exceptional Weyl groups were computed by De Concini and Procesi one year later. We consider the hyperplane arrangements associated with ideals of the root system of a Weyl group. These arrangements were introduced in 2006 by Sommers and Tymoczko. The talk assumes that the subject does not belong to the main field of expertise of a significant part of the audience. That is why enough time will be taken to define important notions like the Tutte polynomial, the Weyl groups, and the ideals of a root system even if they could seem to be basic. We also expose our results from 2020 concerning the Tutte polynomial of hyperplane arrangements associated with ideals of a classical root system. Then, we finish with an introduction of an open problem on the Tutte polynomial of hyperplane arrangements associated with ideals of an exceptional root system.