Poincaré polynomials of generic torus orbit closures in Schubert varieties

Abstract

The closure of a generic torus orbit in the flag variety G / B G/B of type A 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 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.

Torus orbit closures in the flag variety

“…The following is well-known for simple polytopes and is a consequence of discrete Morse theory. Theorem 4.12 (see [52,Theorem 2.7] for example). Let P ⊂ R n be a lattice polytope.…”

“…One can check that the linear function h in Lemma 4.15 satisfies the conditions in Theorem 4.12 so that we obtain the following theorem extending Theorem 3.8. Theorem 4.17 ([52,Theorem 3.6]). Let Y w be the generic torus orbit closure in the Schubert variety X w for w ∈ S n .…”

“…Recently, the authors investigated the topology and geometry of (not necessarily generic) torus orbit closures in connection with combinatorics. In this chapter, we present some of the work by Klyachko and Gelfand-Serganova and our recent works (see [46], [51], [48], [49], [52]) on the topology, geometry, and combinatorics of torus orbit closures in the flag variety. We mainly focus on type A case although some of the arguments work for other Lie types.…”