Linear degenerations of flag varieties

Abstract

Abstract. Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.

The Minkowski Property and Reflexivity of Marked Poset Polytopes

“…Motivated by the work on linear degenerations of flag varieties [7], seeking for intermediate lattice polytopes between the marked order and the marked chain polytopes, as well as toric degenerations of the linear degenerate flag varieties to these polytopes, becomes a meaningful question.…”

Totally nonnegative Grassmannians, Grassmann necklaces, and quiver Grassmannians

“…One may vary the representation U ωn keeping its dimension unchanged. Then one gets a family of quiver Grassmannians in the spirit of [CFFFR17,F12]. It would be interesting to study this family.…”

“…The entries of are all equal to n , and the th entry of equals . In the setting of linear degenerations of the flag variety of type A, the degenerations with rank tuple between the flag and the so-called Feigin degeneration are all of the same dimension [CFFFR17, Theorem A]. The construction of the quiver representation is somehow analogous to the construction of [CFR12, Definition 2.5]. However, despite that, there are linear degenerations of with rank tuple between and which are not of dimension , as explained in the following example. We collect some intermediate degenerations which are not of dimension .…”

A Continuous Family of Marked Poset Polytopes

“…Motivated by the recent work on linear degenerate flag varieties [3], the first two authors introduced marked chain-order polytopes [4], which are mixtures of the two, i.e., for each order ideal of the poset, one imposes chain conditions on the coordinates in the order ideal, and order conditions on the coordinates in its complement. They proved that these marked chain-order polytopes form an Ehrhart equivalent family of lattice polytopes, containing the marked order and marked chain polytopes as extremal cases.…”