On Plücker equations characterizing Grassmann cones

  • Letterio Gatto

    Politecnico di Torino, Italy
  • Parham Salehyan

    IBILCE/UNESP, São José do Rio Preto, Brazil
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The KP hierarchy (after Kadomtsev and Petshiasvily) is a system of infinitely many PDEs in Lax form defining a universal family of iso-spectral deformation of an ordinary linear differential operator. It is a classical result due to Sato's japanese school that the rational solutions to the KP hierarchy are parametrized by a cone over an infinite-dimensional Grassmann variety. The present survey will revisit this fact from the point of view of Schubert derivations on a Grassmann algebra. These enable to encode the classical Plücker equations of Grassmannians of rr-dimensional subspaces in a formula whose limit for rr \to \infty coincides with the KP hierarchy, phrased in terms of vertex operators, showing in particular how the latter is intimately related to Schubert calculus.