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Matroid products in tropical geometry

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Abstract

Symmetric powers of matroids were first introduced by Lovasz (Combinatorial surveys, in: Proceedings 6th British combinatorial conference, pp 45-86, 1977) and Mason (Algebr Methods Graph Theory 1:519-561, 1981) in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor-closed and has infinitely many forbidden minors.

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Acknowledgements

I’d like to give some special thanks to Felipe Rincón and Alex Fink for their valuable insight and guidance with this paper, as well as to Josephine Yu, Diane Maclagan, and Zach Walsh for helpful conversations that inspired multiple improvements to the results in this paper.

Funding

The author is funded by Queen Mary University of London through the Queen Mary Principal’s Award.

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  1. School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK

    Nicholas Anderson

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  1. Nicholas Anderson
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Correspondence to Nicholas Anderson.

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Anderson, N. Matroid products in tropical geometry. Res Math Sci 11, 38 (2024). https://doi.org/10.1007/s40687-024-00452-z

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