Generic torus orbit closures in Schubert varieties

Abstract

The closure of a generic torus orbit in the flag variety G/B of type A is known to be a permutohedral variety and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G/B. When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

Complexity of the usual torus action on Kazhdan-Lusztig varieties

“…We also discuss the relation between σ v,w and the cone of weights of the usual torus action on affine neighborhoods of torus fixed points in Schubert variety X w , i.e. vΩ • id ∩ X w , studied in [LM20]. Finally, in section 4.3 we interpret our results in terms of directed graphs.…”

“…We also explain the relation to the description of the weight cone of the T -action on X w ∩ vΩ o id , investigated in [LM20]. 4.2.3.…”

The Smooth Torus Orbit Closures in the Grassmannians

“…The first section is devoted to the proof of Theorem B above. To prove it, we associate a graph to each vertex v of a matroid polytope by using the edge vectors emanating from the vertex v. This graph is an analogue of that introduced in [8]. The key fact we use is that the v is a simple vertex if and only if the graph has no cycle.…”

“…The following lemma can easily be proved, see [8,Section 6]. For a vertex v = δ J ∈ ∆ n,k and a positive integer s we introduce two notations:…”

Generic Torus Orbit Closures in Flag Bott Manifolds

“…Generic torus orbit closures in a generalized flag manifold G/P are studied in [FH91] and [Dab96], and arbitrary orbit closures in Grassmannian manifolds are studied in [GGMS87], [GS87], [BT18]. Furthermore generic torus orbit closures in Schubert varieties are studied in [LM18].…”