Linear embeddings of grassmannians and ind-grassmannians

By a grassmannian we understand a usual complex grassmannian or possibly an orthogonal or symplectic grassmannian. We classify, with few exceptions, linear embeddings of grassmannians into larger grassmannians, where the linearity requirement is the condition that the embedding induces an isomorphism on Picard groups. This classification implies that most linear embeddings of grassmannians are equivariant. A linear ind-grassmannian is the direct limit of a chain of linear embeddings of grassmannians. We conclude the paper by classifying linear embeddings of linear ind-grassmannians.