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Newton-Okounkov bodies and cluster duality for Grassmannians

Connections for Women: Enumerative Geometry Beyond Numbers January 18, 2018 - January 19, 2018

January 19, 2018 (02:00 PM PST - 03:00 PM PST)
Speaker(s): Lauren Williams (Harvard University)
Location: SLMath: Eisenbud Auditorium
  • mirror symmetry

  • Grassmannian

  • cluster algebra

  • cluster variety

  • Newton-Okounkov body

  • Schubert calculus

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification



We use the A and X-cluster structures on the Grassmannian to exhibit a polytopal manifestation of mirror symmetry for Grassmannians. From a given cluster seed we obtain both an X-cluster chart and an A-cluster chart for the Grassmannian. We use the X-cluster chart and a naturally defined valuation to construct a Newton-Okounkov body, defined as the convex hull of points. Meanwhile we use the A-cluster to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a polytope, defined by inequalities. We prove that the Newton-Okounkov body and the superpotential polytope coincide. In the case that our A-cluster consists of Plucker coordinates, we also give a formula for each lattice point of these polytopes in terms of Young diagrams; using a result of Fulton-Woodward, this formula has an interpretation in terms of quantum cohomology. This is joint work with Konstanze Rietsch.

30494?type=thumb Williams Notes 236 KB application/pdf Download
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