$G$-Tutte polynomials and abelian Lie group arrangements

TL;DR

This paper introduces the $G$-Tutte polynomial for abelian group arrangements, generalizing several known polynomials and capturing topological and enumerative invariants of these arrangements.

Contribution

It defines the $G$-Tutte polynomial for abelian group arrangements, unifying various existing polynomials and exploring their properties and differences.

Findings

$G$-Tutte polynomial generalizes multiple known polynomials.

It encodes topological and enumerative invariants.

Differences between arithmetic Tutte and $G$-Tutte polynomials are discussed.

Abstract

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, Br\"and\'en-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 Poincar\'e 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.