Toric Bruhat interval polytopes

Abstract

For two elements v and w of the symmetric group S-n with v <= w in Bruhat order, the Bruhat interval polytope Q(v,w) is the convex hull of the points (z(1), ..., z(n)) is an element of R-n with v <= z <= w. It is known that the Bruhat interval polytope Q(v,w) is the moment map image of the Richardson variety X-w-1(v-1). We say that Q(v,w) is toricif the corresponding Richardson variety X-w-1(v-1) is a toric variety. We show that when Q(v,w) is toric, its combinatorial type is determined by the poset structure of the Bruhat interval [v, w] while this is not true unless Q(v,w) is toric. We are concerned with the problem of when Q(v,w) is (combinatorially equivalent to) a cube because Q(v,w) is a cube if and only if X-w-1(v-1) is a smooth toric variety. We show that a Bruhat interval polytope Q(v,w) is a cube if and only if Q(v,w) is toric and the Bruhat interval [v, w] is a Boolean algebra. We also give several sufficient conditions on vand wfor Q(v,w) to be a cube. (C) 2020 Elsevier Inc. All rights reserved.