Toric Bruhat interval polytopes

Abstract

For two elements v and w of the symmetric group Sn with v≤w in Bruhat order, the Bruhat interval polytope Qv,w is the convex hull of the points (z(1),…,z(n))∈Rn with v≤z≤w. It is known that the Bruhat interval polytope Qv,w is the moment map image of the Richardson variety Xw−1v−1. We say that Qv,w is toric if the corresponding Richardson variety Xw−1v−1 is a toric variety. We show that when Qv,w is toric, its combinatorial type is determined by the poset structure of the Bruhat interval [v,w] while this is not true unless Qv,w is toric. We are concerned with the problem of when Qv,w is (combinatorially equivalent to) a cube because Qv,w is a cube if and only if Xw−1v−1 is a smooth toric variety. We show that a Bruhat interval polytope Qv,w is a cube if and only if Qv,w is toric and the Bruhat interval [v,w] is a Boolean algebra. We also give several sufficient conditions on v and w for Qv,w to be a cube.