Automorphisms of Grassmannians

Abstract

For a complex vector space V \mathcal {V} of dimension n n , the group of holomorphic automorphisms of the Grassmannian Gr ⁡ ( p , V ) \operatorname {Gr}(p,\mathcal {V}) can be identified with the subgroup of P G 1 ( ∧ p V ) {\mathbf {P}}G1({\wedge ^p}\mathcal {V}) preserving the Grassmannian. Using this, Chow showed Aut ⁡ (Gr ( p , V ) ) = P Gl ⁡ ( V ) \operatorname {Aut} {\text {(Gr}}(p,\mathcal {V})) = {\mathbf {P}}\operatorname {Gl} (\mathcal {V}) for n ≠ 2 p n \ne 2p , and P G 1 ( V ) {\mathbf {P}}G1(\mathcal {V}) is a normal subgroup of index 2 in A u t ( G r ⁡ ( p , V ) ) \operatorname {Aut(Gr}(p,\mathcal {V})) for n = 2 p n = 2p . We prove a version of Chow’s result for a separable Hilbert space H \mathcal {H} . Theorem. P Gl ⁡ ( H ) {\mathbf {P}}\operatorname {Gl} (\mathcal {H}) is the subgroup of P Gl ⁡ ( ∧ p H ) {\mathbf {P}}\operatorname {Gl} ({\wedge ^p}\mathcal {H}) which preserves Gr ⁡ ( p , H ) \operatorname {Gr}(p,\mathcal {H}) . That is, if R R is an invertible linear operator on ∧ p H {\wedge ^p}\mathcal {H} which preserves decomposable p p -vectors, then there exists S S , an invertible linear operator on H \mathcal {H} , such that R = ∧ p S R = {\wedge ^p}S .