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Multi-splits and Tropical Linear Spaces from Nested Matroids

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Abstract

We present an explicit combinatorial description of a special class of facets of the secondary polytopes of hypersimplices. These facets correspond to polytopal subdivisions called multi-splits. We show that the maximal cells in a multi-split of a hypersimplex are matroid polytopes of nested matroids. Moreover, we derive a description of all multi-splits of a product of simplices. Additionally, we present a computational result to derive explicit lower bounds on the number of facets of secondary polytopes of hypersimplices.

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Acknowledgements

I am indebted to Michael Joswig and Georg Loho for various helpful suggestions and I thank Alex Fink and the three anonymous referees for their comments. My research is carried out in the framework of Matheon supported by Einstein Foundation Berlin (Project “MI6 - Geometry of Equilibria for Shortest Path”). Furthermore, I thank Institut Mittag-Leffler for the hospitality during my stay in early 2018.

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  1. Institut für Mathematik, TU Berlin, Str. des 17 Juni 136, 10623, Berlin, Germany

    Benjamin Schröter

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  1. Benjamin Schröter
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Schröter, B. Multi-splits and Tropical Linear Spaces from Nested Matroids. Discrete Comput Geom 61, 661–685 (2019). https://doi.org/10.1007/s00454-018-0021-1

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