Linear embeddings of grassmannians and ind-grassmannians
Ivan Penkov, Valdemar Tsanov
TL;DR
The paper provides a comprehensive classification of linear embeddings between grassmannians and, extending to ind-grassmannians, of linear ind-embeddings. Building on Penkov–Tikhomirov–Lin-ind-Grass, it shows that most embeddings are standard extensions or factor through projective spaces or quadrics, with explicit treatment of cross-type and spinor cases. It also proves that, aside from a few exceptional configurations, linear embeddings are equivariant for appropriate classical or Levi subgroups, elucidating the homogeneous-structure aspects of these embeddings. The results yield a near-complete picture of how Grassmannians sit inside one another under linear plugs, and they extend to ind-Grassmannians, connecting to Sato Grassmannians and infinite-dimensional homogeneous spaces with implications for representation theory and algebraic geometry of ind-varieties.
Abstract
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.