Petter Brändén: Toric arrangements and multivariate arithmetic Tutte polynomials

Toric arrangements, i.e., arrangements of suitable hyper-surfaces in a torus, are not as well understood as hyperplane arrangements. To remedy this the notion of an arithmetic matroid and arithmetic Tutte polynomial have been introduced as toric counterparts of matroids and Tutte polynomials. In this talk we will study a multivariate arithmetic Tutte polynomial, and prove that it encodes combinatorial, algebraic and geometric properties of the toric arrangement.  A classical result of Fortuin and Kasteleyn shows that the multivariate Tutte polynomial of a graphic matroid is the partition function of the q-state Potts model. To any list of vectors with integer coordinates we associate a generalized Potts model for which the partition function is a quasi-polynomial that restricts to the ordinary as well as to the arithmetic multivariate Tutte polynomial. This can be seen as a generalization of the "finite field method" and provides a Fortuin-Kasteleyn representation of the multivariate (arithmetic) Tutte polynomial of any matroid representable over the rational numbers.  This is joint work with Luca Moci.