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Representations of Weakly Multiplicative Arithmetic Matroids are Unique

  • Matthias LenzEmail author
Article
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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 integral cohomology ring of a centered toric arrangement whose arithmetic matroid is weakly multiplicative is determined by its poset of layers. This partially answers a question asked by Callegaro–Delucchi.

Keywords

Arithmetic matroid Representation Toric arrangement Combinatorial topology 

Mathematics Subject Classification

Primary 05B35 52C35 Secondary 14M15 14N20 57N65 

Notes

Acknowledgements

The author would like to thank Elia Saini for several interesting discussions and an anonymous referee for many helpful suggestions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de FribourgFribourgSwitzerland

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