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H 1 Newton-Okounkov bodies for Grassmannians

Auteurs : Williams, Lauren (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : In joint work with Konstanze Rietsch (arXiv:1712.00447), we use the $\mathcal{X}$-cluster structure on the Grassmannian and the combinatorics of plabic graphs to associate a Newton-Okounkov body to each $\mathcal{X}$-cluster. This gives, for each $\mathcal{X}$-cluster, a toric degeneration of the Grassmannian. We also describe the Newton-Okounkov bodies quite explicitly: we show that their facets can be read off from $\mathcal{A}$-cluster expansions of the superpotential. And we give a combinatorial formula for the lattice points of the Newton-Okounkov bodies, which has a surprising interpretation in terms of quantum Schubert calculus.

    Keywords : Grassmannian; plabic graph; cluster chart; network chart; Newton-Okounkov body; $\mathcal{X}$-cluster; toric degeneration; $\mathcal{A}$-cluster

    Codes MSC :
    05E10 - Combinatorial aspects of representation theory
    14M15 - Grassmannians, Schubert varieties, flag manifolds
    14M25 - Toric varieties, Newton polyhedra
    14M27 - Compactifications; symmetric and spherical varieties

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 28/03/2018
      Date de captation : 21/03/2018
      Collection : Research talks ; Combinatorics ; Algebraic and Complex Geometry
      Format : MP4
      Durée : 00:51:51
      Domaine : Algebraic & Complex Geometry ; Combinatorics
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2018-03-21_L_Williams.mp4

    Informations sur la rencontre

    Nom de la rencontre : Cluster algebras: twenty years on / Vingt ans d'algèbres amassées
    Organisateurs de la rencontre : Amiot, Claire ; Baur, Karin ; Dupont, Grégoire ; Marsh, Robert J. ; Morier-Genoud, Sophie ; Palu, Yann ; Pilaud, Vincent ; Plamondon, Pierre-Guy ; Qin, Fan
    Dates : 19/03/2018 - 23/03/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1777.html

    Citation Data

    DOI : 10.24350/CIRM.V.19384003
    Cite this video as: Williams, Lauren (2018). Newton-Okounkov bodies for Grassmannians. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19384003
    URI : http://dx.doi.org/10.24350/CIRM.V.19384003


    Voir aussi

    Bibliographie

    1. Rietsch, K., & Williams, L. (2017). Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians. - https://arxiv.org/abs/1712.00447

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