Injectivity of generalized Wronski maps

TL;DR

This paper characterizes when linear projections on Plücker space induce branched covers on Grassmannians, showing that such maps are essentially Lagrangian involutions in symplectic spaces, with applications to Wronski maps and pole placement.

Contribution

It proves that automorphisms preserving fibers of these projections are Lagrangian involutions, revealing the structure of such maps in symplectic geometry.

Findings

Grassmannian automorphisms preserving fibers are Lagrangian involutions

Characterization of projections as symplectic involutions

Applications to Wronski maps and pole placement maps

Abstract

We study linear projections on Pluecker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of m-dimensional linear subspaces in a symplectic vector space of dimension 2m, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples.