The Jacobian of a Sixth-Root-of-Unity Matroid

TL;DR

This paper introduces the Jacobian group for sixth-root-of-unity matroids, extending the concept from regular matroids and establishing foundational properties for this new class.

Contribution

It defines the Jacobian group for sixth-root-of-unity matroids and explores their basic properties, broadening the understanding of matroid invariants.

Findings

Defined the Jacobian group for sixth-root-of-unity matroids

Established fundamental properties of these Jacobian groups

Extended concepts from regular to complex unimodular matroids

Abstract

The Jacobian group (also called the sandpile group, Picard group, or critical group) of a graph or, more generally, of a regular matroid has been well studied. Sixth-root-of-unity matroids, also called complex unimodular matroids, are generalizations of regular matroids. This paper provides a definition, and establishes some basic properties, of the Jacobian group of a sixth-root-of-unity matroid.