G-Tutte Polynomials and Abelian Lie Group Arrangements

Abstract

AbstractFor a list $\mathcal{A}$ of elements in a finitely generated abelian group $\Gamma $ and an abelian group $G$, we introduce and study an associated $G$-Tutte polynomial, 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, Brändén–Moci’s 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 Poincaré 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.