Combinatorial mutations and block diagonal polytopes

Oliver Clarke ^[1] ; Akihiro Higashitani ^[2] ; Fatemeh Mohammadi ^[3]
1. [1] University of Bristol
  
  University of Bristol
  
  Reino Unido
2. [2] Osaka University
  
  Osaka University
  
  Kita Ku, Japón
3. [3] Ghent University
  
  Ghent University
  
  Arrondissement Gent, Bélgica
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Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 2, 2022, págs. 305-335
Idioma: inglés
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Resumen
- 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.