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We introduce and study the notion of the $G$-Tutte polynomial for a list $mathcal{A}$ of elements in a finitely generated abelian group $Gamma$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite abelian groups to $G$. The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Branden-Mocis arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial for a finite CW-complex. As in the classical case, $G$-Tutte polynomials carry topological and enumerative information (e.g., the Euler characteristic, point counting and the Poincare polynomial) of abelian Lie group arrangements. We also discuss differences between the arithmetic Tutte and the $G$-Tutte polynomials related to the axioms for arithmetic matroids and the (non-)positivity of coefficients.
The purpose of this paper is twofold. Firstly, we generalize the notion of characteristic polynomials of hyperplane and toric arrangements to those of certain abelian Lie group arrangements. Secondly, we give two interpretations for the chromatic qua
We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollobas and Riordans ribbon graph polynomial, $R(G)$, and the top
The braid arrangement is the Coxeter arrangement of the type $A_ell$. The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations. In this paper, we give an explic
Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic quasi-polynomial. In this
The (extended) Linial arrangement $mathcal{L}_{Phi}^m$ is a certain finite truncation of the affine Weyl arrangement of a root system $Phi$ with a parameter $m$. Postnikov and Stanley conjectured that all roots of the characteristic polynomial of $ma