Generic torus orbit closures in Schubert varieties

Abstract

The closure of a generic torus orbit in the flag variety G/B of type An−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 Xw(w∈Sn) and study its closure Yw. We identify the maximal cone in the fan of Yw corresponding to a fixed point uB(u≤w), associate a graph Γw(u) to each u≤w, and show that Yw is smooth at uB if and only if Γw(u) is a forest. We also introduce a polynomial Aw(t) for each w, which agrees with the Eulerian polynomial when w is the longest element of Sn, and show that the Poincaré polynomial of Yw agrees with Aw(t2) when Yw is smooth.