An algebraic-geometric construction of ind-varieties of generalized flags

TL;DR

This paper introduces a purely algebraic-geometric method to construct ind-varieties of generalized flags by characterizing admissible linear embeddings of flag varieties and showing their limits form these ind-varieties.

Contribution

It provides an explicit algebraic-geometric construction of ind-varieties of generalized flags, connecting linear algebra with algebraic geometry in the context of ind-groups.

Findings

Admissible linear embeddings have an explicit linear algebraic form.

Limits of admissible embeddings form ind-varieties of generalized flags.

The construction aligns with previous definitions using ind-groups like SL(∞).

Abstract

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags as defined in [DP]. These latter ind-varieties have been introduced in terms of the ind-group SL(\infty) (respectively, O(\infty) or Sp(\infty) for isotropic generalized flags), and the current paper constructs them in purely algebraic-geometric terms