Tutte Polynomials of arrangements of ideal type
Final Report Abstract
Tutte polynomials are some of the most-studied invariants of graphs. This two-variable polynomial encodes substantial information of the combinatorics of a graph. It specializes to several important graph polynomials (including the chromatic, flow and reliability polynomials). Significant features of the Tutte polynomial have also been shown in diverse areas of mathematics and physics. For instance, it appears as the Jones and homfly polynomials in knot theory, and as the Ising and Potts model partition functions in statistical mechanics. The Tutte polynomial is an important invariant in the theory of graphs, matroids and arrangements alike. There is a strong connection between these three notions, where the one in the matroid setting is the most general one. The theory of Tutte polynomials is a very active field of research. Originally, Tutte was interested in the colorings of graphs, in particular in the chromatic polynomial. It is a specialization of the Tutte polynomial for graphs, which he defined in 1954. The generalization to matroids was quite natural and first formulated by Crapo in 1969. In 2007, Ardila introduced the Tutte polynomial of a hyperplane arrangement. A matroid isomorphism invariant is a function that only depends on the isomorphism class of a matroid. A so called T-G invariant in addition fulfills certain recursions. Emphasizing its significance, it turns out that every T-G invariant is in fact an evaluation of the Tutte polynomial. Therefore, other such invariants, like e.g. the characteristic polynomial, the number of bases or the number of independent sets, can be derived from the Tutte polynomial of a matroid. Both, the Tutte polynomial as well as the characteristic polynomial of an arrangement agree with the corresponding polynomial of the matroid associated to the arrangement. While there are closed formulas for the Tutte polynomials of real reflection arrangements A(W ), those of ideal arrangements have not been studied thus far. Schauenburg made a first step towards that direction by providing an algorithm that uses matroid theoretic means to compute the Tutte polynomial associated to an ideal arrangement AI . The aim of this project was then to provide an explicit algorithm that computes the Tutte polynomial of a given arrangement of ideal type AI based on the theory of matroids. Here the initial data should simply be the antichain of roots generating the ideal I in the set of positive roots of the underlying root system. While Coxeter arrangements A(W ) are well understood, this is not the case for their subarrangements in general. For instance, Tutte determined the (generating function) of the Tutte polynomial for the braid arrangement and Ardila calculated the Tutte polynomial for each of the remaining classical Coxeter arrangements. One aim was to show that a certain combinatorial property gives an inductive method to determine the Tutte polynomial of a large class of the arrangements of ideal type AI . This inductive technique was used to show that many of the o arrangements AI are inductively free. By iterating this inductive method, we had hoped to obtain closed formulas for the Tutte polynomial of an arrangement of ideal type in most classical instances. Unfortunately, this was too ambitious and did not turn out to be a feasible undertaking in the end.
Publications
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Arrangements of ideal type are inductively free, Internat. J. Algebra Comput. 29 (2019), no. 5, 761–773
M. Cuntz, G. Röhrle, and A. Schauenburg
