# Representations of weakly multiplicative arithmetic matroids are unique

## TL;DR

This paper proves that weakly multiplicative arithmetic matroids represented by integer matrices have unique representations, linking their structure to the cohomology ring of associated toric arrangements and answering a question in the field.

## Contribution

It establishes the uniqueness of integer matrix representations for weakly multiplicative arithmetic matroids and connects this to the cohomology of toric arrangements.

## Key findings

- Integer matrix representations of weakly multiplicative arithmetic matroids are unique.
- The cohomology ring of centered toric arrangements is determined by the poset of layers.
- Partially answers a question by Callegaro-Delucchi regarding these structures.

## Abstract

An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer matrix, then this matrix is uniquely determined. This implies that the integer cohomology ring of a centred toric arrangement whose arithmetic matroid is weakly multiplicative is determined by its poset of layers. This partially answers a question asked by Callegaro-Delucchi.