Automorphisms of Grassmannians

Abstract:

For a complex vector space $\mathcal {V}$ of dimension $n$, the group of holomorphic automorphisms of the Grassmannian $\operatorname {Gr}(p,\mathcal {V})$ can be identified with the subgroup of ${\mathbf {P}}G1({\wedge ^p}\mathcal {V})$ preserving the Grassmannian. Using this, Chow showed $\operatorname {Aut} {\text {(Gr}}(p,\mathcal {V})) = {\mathbf {P}}\operatorname {Gl} (\mathcal {V})$ for $n \ne 2p$, and ${\mathbf {P}}G1(\mathcal {V})$ is a normal subgroup of index 2 in $\operatorname {Aut(Gr}(p,\mathcal {V}))$ for $n = 2p$. We prove a version of Chow’s result for a separable Hilbert space $\mathcal {H}$. Theorem. ${\mathbf {P}}\operatorname {Gl} (\mathcal {H})$ is the subgroup of ${\mathbf {P}}\operatorname {Gl} ({\wedge ^p}\mathcal {H})$ which preserves $\operatorname {Gr}(p,\mathcal {H})$. That is, if $R$ is an invertible linear operator on ${\wedge ^p}\mathcal {H}$ which preserves decomposable $p$-vectors, then there exists $S$, an invertible linear operator on $\mathcal {H}$, such that $R = {\wedge ^p}S$.