Bruhat interval polytopes

Let u and v be permutations on n letters, with u≤v in Bruhat order. A Bruhat interval polytope Qu,v is the convex hull of all permutation vectors z=(z(1),z(2),…,z(n)) with u≤z≤v. Note that when u=e and v=w0 are the shortest and longest elements of the symmetric group, Qe,w0 is the classical permutohedron. Bruhat interval polytopes were studied recently in [15] by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety.In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and R-polynomials, we also give a generalization of the standard recurrence for R-polynomials. Finally, we define a more general class of polytopes called Bruhat interval polytopes for G/P, which are moment map images of (closures of) totally positive cells in (G/P)≥0, and are a special class of Coxeter matroid polytopes. Using tools from total positivity and the Gelfand-Serganova stratification, we show that the face of any Bruhat interval polytope for G/P is again a Bruhat interval polytope for G/P.