Morse theoretic aspects of Plücker embeddings

Abstract

Plücker embeddings of real and complex Grassmannians ${Gr_p({\boldsymbol F}^n)}$ (${\boldsymbol F} = {\boldsymbol R}$, ${\boldsymbol C}$) and manifolds associated to Plücker embeddings are analyzed from the viewpoint of critical point theory. It is proved that the Plücker embedding of ${Gr_p({\boldsymbol F}^n)}$ with $1\le p\le n-p$ is taut if and only if $1\le p\le 2$, in contrast to standard embeddings which are known to be taut for all $1\le p\le n$. As an application, we show a convexity property of Plücker embeddings, which is an analogue to Schur-Horn convexity on Hermitian matrices.