Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Automorphisms of Grassmannians


Author: Michael J. Cowen
Journal: Proc. Amer. Math. Soc. 106 (1989), 99-106
MSC: Primary 14M15; Secondary 32M10
MathSciNet review: 938909
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 14M15, 32M10

Retrieve articles in all journals with MSC: 14M15, 32M10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0938909-8
PII: S 0002-9939(1989)0938909-8
Article copyright: © Copyright 1989 American Mathematical Society