On Plücker equations characterizing Grassmann cones

Abstract

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 $r$-dimensional subspaces in a formula whose limit for $r\to \infty$ coincides with the KP hierarchy, phrased in terms of vertex operators, showing in particular how the latter is intimately related to Schubert calculus.