Generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety G/B of type A(n-1) is known to be a permutohedral variety and well studied. In this paper we introduce the notion of a generic torus orbit in the Schubert variety X-w (w is an element of S-n) and study its closure Y-w. We identify the maximal cone in the fan of Y-w corresponding to a fixed point uB (u <= w), associate a graph Gamma(w) (u) to each u <= w, and show that Y-w is smooth at uB if and only if Gamma(w) (u) is a forest. We also introduce a polynomial A(w)(t) for each w, which agrees with the Eulerian polynomial when w is the longest element of S-n, and show that the Poincare polynomial of Y-w agrees with A(w)(t(2)) when Y-w is smooth. (C) 2019 Elsevier Inc. All rights reserved.