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Representations of weakly multiplicative arithmetic matroids are unique
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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.
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Representations of torsion-free arithmetic matroids
A new reduction and signed Hermite normal form let every representation of a torsion-free arithmetic matroid be computed up to equivalence, yielding a sharpened upper bound and counterexamples to two shellability conjectures.
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