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

     

  Mostrar afiliaciones +
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 2, 2022, págs. 305-335
• Idioma: inglés
• DOI: 10.1007/s13348-021-00321-w
• Enlaces
  • Texto completo
• 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.

• Referencias bibliográficas
  • Akhtar, M., Coates, T., Galkin, S., Kasprzyk, A.M.: Minkowski polynomials and mutations. SIGMA. Symm. Integr. Geometry: Methods Appl. 8, 094...
  • Alexandersson, P.: Gelfand-Tsetlin polytopes and the integer decomposition property. Eur. J. Combinat. 54, 1–20 (2016)
  • An, B.H., Cho, Y., Kim, J.S.: On the f-vectors of Gelfand-Tsetlin polytopes. Eur. J. Combinat. 67, 61–77 (2018)
  • Anderson, D.: Okounkov bodies and toric degenerations. Math. Annalen 356(3), 1183–1202 (2013)
  • Bigatti, A.M., La Scala, R., Robbiano, L.: Computing toric ideals. J. Symbol. Comput. 27(4), 351–365 (1999)
  • Bossinger, L., Fang, X., Fourier, G., Hering, M., Lanini, M.: Toric degenerations of Gr(2, n) and Gr(3,6) via plabic graphs. Ann. Combinat....
  • Bossinger, L., Lamboglia, S., Mincheva, K., Mohammadi, F.: Computing toric degenerations of flag varieties. In: Smith, G., Sturmfels, B. (eds.)...
  • Bossinger, L., Mohammadi, F., Chávez, A.N.: Families of Gröbner degenerations, Grassmannians and universal cluster algebras. arXiv preprintarXiv:2007.14972,...
  • Chary Bonala, N., Clarke, O., Mohammadi, F.: Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians...
  • Chen, T., Mehta, D.: Parallel degree computation for binomial systems. J. Symbol. Comput. 79, 535–558 (2017)
  • Clarke, O., Mohammadi, F.: Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux. J. Algebra 559, 646–678...
  • Clarke, O., Mohammadi, F.: Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux. J. Symbol....
  • Clarke, O., Mohammadi, F.: Toric degenerations of flag varieties from matching field tableaux. J. Pure Appl. Algebra 225(8), 106624 (2021)
  • Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. American Mathematical Society (2011)
  • Escobar, L., Harada, M.: Wall-crossing for Newton-Okounkov bodies and the tropical Grassmannian. Int. Math. Res. Notices, rnaa230, (2020)
  • Fink, A., Rincón, F.: Stiefel tropical linear spaces. J. Combinat. Theory, Ser A 135, 291–331 (2015)
  • Fujita, N., Higashitani, A.: Newton-okounkov bodies of flag varieties and combinatorial mutations. International Mathematics Research Notices,...
  • Galkin, S., Usnich, A.V.: Mutations of potentials. Preprint IPMU 10–0100, (2010)
  • Gawrilow, E., Joswig, M.: polymake: a framework for analyzing convex polytopes. In: Polytopes—combinatorics and computation (Oberwolfach,...
  • Herrmann, S., Jensen, A., Joswig, M., Sturmfels, B.: How to draw tropical planes. Electron. J. Combinat. 16(2), R6 (2009)
  • Higashitani, A.: Two poset polytopes are mutation-equivalent. arXiv:2002.01364, (2020)
  • Kahle, T.: Decompositions of binomial ideals. Ann. Institu. Statist. Math. 62(4), 727–745 (2010)
  • Kaveh, K., Manon, C.: Khovanskii bases, higher rank valuations, and tropical geometry. SIAM J. Appl. Algebra Geom. 3(2), 292–336 (2019)
  • Kogan, M., Miller, E.: Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes. Adv. Math. 193(1), 1–17 (2005)
  • Loho, G., Smith, B.: Matching fields and lattice points of simplices. Adv. Math. 370, 107232 (2020)
  • Mohammadi, F., Shaw, K.: Toric degenerations of Grassmannians from matching fields. Algebraic Combinat. 2(6), 1109–1124 (2019)
  • Rietsch, K., Williams, L.: Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians. Duke Math. J. 168(18), 3437–3527...
  • Robbiano, L., Sweedler, M.: Subalgebra bases. In: Bruns, W., Simis, A. (eds.) Commutative Algebra, pp. 61–87. Springer (1990)
  • Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)
  • Sturmfels, B.: Gröbner Bases and Convex Polytopes. American Mathematical Society (1996)
  • Sturmfels, B., Zelevinsky, A.: Maximal minors and their leading terms. Adv. Math. 98(1), 65–112 (1993)