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# Injectivity of Generalized Wronski Maps

Published:2017-03-16
Printed: Dec 2017
• Yanhe Huang,
Department of Mathematics, University of California , Berkeley, CA 94720, USA
• Frank Sottile,
Department of Mathematics, Texas A&M University , College Station, Texas 77843, USA
• Igor Zelenko,
Department of Mathematics, Texas A&M University , College Station, Texas 77843, USA
 Format: LaTeX MathJax PDF

## Abstract

We study linear projections on Plücker 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.
 Keywords: Wronski map, Plücker embedding, curves in Lagrangian Grassmannian, self-adjoint linear differential operator, symmetric linear control system, pole placement map
 MSC Classifications: 14M15 - Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 34A30 - Linear equations and systems, general 93B55 - Pole and zero placement problems

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