Bruhat Interval Polytopes

TL;DR

This paper explores the combinatorial structure of Bruhat interval polytopes, providing inequalities, dimension formulas, and face characterizations, and extends the concept to G/P spaces with connections to total positivity and Coxeter matroids.

Contribution

It introduces new combinatorial descriptions, face properties, and generalizations of Bruhat interval polytopes, including for G/P spaces, enhancing understanding of their geometric and algebraic structure.

Findings

Every face of a Bruhat interval polytope is itself a Bruhat interval polytope.

Provides an inequality description and a dimension formula for Bruhat interval polytopes.

Generalizes the concept to Bruhat interval polytopes for G/P, linking to total positivity and Coxeter matroids.

Abstract

Let u and v be permutations on n letters, with u <= v in Bruhat order. A Bruhat interval polytope Q_{u,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=w_0 are the shortest and longest elements of the symmetric group, Q_{e,w_0} is the classical permutohedron. Bruhat interval polytopes were studied recently 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 the non-negative part of G/P, 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.