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Combinatorial mutations and block diagonal polytopes

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Abstract

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes, and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover, the polytopes arising through mutations are Newton-Okounkov bodies for the Grassmannians with respect to certain full-rank valuations. We produce a large family of such polytopes, extending the family of so-called block diagonal matching fields.

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Acknowledgements

OC and FM would like to thank the organizers of the “Workshop on Commutative Algebra and Lattice Polytopes” at RIMS in Kyoto, where this work began. OC is supported by EPSRC Doctoral Training Partnership award EP/N509619/1. AH is partially supported by JSPS KAKENHI \(\sharp\)20K03513. FM was partially supported by EPSRC Fellowship EP/R023379/1, the BOF Starting Grant BOF/STA/201909/038 of Ghent University, and FWO grant G023721N.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Oliver Clarke

  2. Department of Pure and Applied Mathematics, Osaka University, Suita, Osaka, 565-0871, Japan

    Akihiro Higashitani

  3. Department of Mathematics: Algebra and Geometry, Ghent University, 9000, Gent, Belgium

    Fatemeh Mohammadi

  4. Department of Mathematics and Statistics, The Arctic University of Norway, 9037, Tromsø, Norway

    Fatemeh Mohammadi

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  1. Oliver Clarke
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  2. Akihiro Higashitani
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  3. Fatemeh Mohammadi
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Correspondence to Fatemeh Mohammadi.

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Clarke, O., Higashitani, A. & Mohammadi, F. Combinatorial mutations and block diagonal polytopes. Collect. Math. 73, 305–335 (2022). https://doi.org/10.1007/s13348-021-00321-w

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