On Pl\"ucker Equations Characterizing Grassmann Cones

TL;DR

This paper connects classical Pl"ucker equations of finite-dimensional Grassmannians with the KP hierarchy in infinite dimensions, using Schubert derivations to encode these relations and reveal their limiting behavior.

Contribution

It introduces a novel encoding of Pl"ucker equations via Schubert derivations that converges to the KP hierarchy in the infinite-dimensional limit.

Findings

Pl"ucker equations are encoded using Schubert derivations.

The limit of these equations as r approaches infinity matches the KP hierarchy.

Provides a new algebraic perspective on the relationship between Grassmannians and integrable systems.

Abstract

Polynomial solutions to the KP hierarchy are known to be parametrized by a cone over an infinite-dimensional Grassmann variety. Using the notion of Schubert derivation on a Grassmann algebra, we encode the classical Pl\"ucker equations of Grassmannians of r-dimensional subspaces in a formula whose limit for $r\rightarrow\infty$ coincides with the KP hierarchy phrased in terms of vertex operators.