On algebro-geometric quotients of torus-invariant subvarieties of the flag variety

Abstract

In this paper, we study the subvarieties of a complex flag variety that are invariant under the action of a maximal torus. Using combinatorial techniques derived from matroid theory, we introduce a decomposition of this variety into affine, locally closed subsets, which we refer to as thin Schubert cells, each indexed by an element of a Cartesian product of matroids. We also show that the set of orbits for each of these thin Schubert cells under the action of the torus is, in fact, an orbit space, which gives rise to topologically trivial fiber bundles. As a consequence of this, we prove the existence of algebro-geometric quotients in the sense of Mumford's Geometric Invariant Theory. The main result of this work is the existence of a surjective map from the set of geometric quotients of thin Schubert cells to the invariant scheme-theoretic points of a complex flag variety, which allows us to decompose it in terms of such quotients and also define a map from the set of these geometric quotients to the invariant homology of the variety. Finally, we give a complete description of the thin Schubert cells for the special case of the flag variety $\mathds{F}_{1<n-1}(\mathds{C}^n)$ and derive explicit counting formulas.