Quiver-Grassmannian realizations of all Schubert varieties

Determine whether every Schubert variety can be identified with a quiver Grassmannian for an enhanced equioriented type A quiver, with enhancements at vertices corresponding to occurrences of the pattern 312 and with the dimension vector recording the associated enhanced subsegments, so that the induced actions yield GKM-variety structures.

Background

Appendix A constructs a quiver-Grassmannian realization and GKM-variety structure for a Schubert variety associated with the permutation 3124. The remark proposes a general mechanism for handling permutations containing the pattern 312 by enhancing the equioriented type A quiver and its representation. The unresolved conjecture is that this construction extends to all Schubert varieties.

References

Conjecturally, all Schubert varieties can be identified with quiver Grassmannians for an enhanced type A quiver. If the permutation avoids the pattern(312), no enhancements are required. For every (312) pattern in the permutation, we have to enhance Q (the equioriented type A quiver) at the vertex i corresponding to the one in the pattern. In the representation M = VQ ⊕ * * * ⊕ VQ we have to enhance all segments above the subsegment of the permutation starting at the i-th vertex. The new entries of the dimension vector e′ count the enhanced subsegments corresponding to the permutation. The C∗- and T -action are analogous to the above example and induce a GKM-variety structure.

— Skeletal Torus Actions and GKM Structures on Quiver Grassmannians of String Representations  (2503.08233 - Pütz, 11 Mar 2025) in Remark A.2, Appendix A, page 14