A positive Grassmannian analogue of the permutohedron

TL;DR

This paper introduces bridge polytopes as a positive Grassmannian analogue of permutohedra, linking combinatorial structures of plabic graphs with geometric properties of polytopes and Bruhat order.

Contribution

It defines bridge polytopes and demonstrates their role in encoding BCFW bridge decompositions and local moves in plabic graphs, extending permutohedron properties to the positive Grassmannian context.

Findings

Bridge polytopes encode BCFW bridge decompositions.

Two-dimensional faces correspond to local moves connecting plabic graphs.

Results generalize to positive parts of Schubert cells.

Abstract

The classical permutohedron Perm is the convex hull of the points (w(1),...,w(n)) in R^n where w ranges over all permutations in the symmetric group. This polytope has many beautiful properties -- for example it provides a way to visualize the weak Bruhat order: if we orient the permutohedron so that the longest permutation w_0 is at the "top" and the identity e is at the "bottom," then the one-skeleton of Perm is the Hasse diagram of the weak Bruhat order. Equivalently, the paths from e to w_0 along the edges of Perm are in bijection with the reduced decompositions of w_0. Moreover, the two-dimensional faces of the permutohedron correspond to braid and commuting moves, which by the Tits Lemma, connect any two reduced expressions of w_0. In this note we introduce some polytopes Br(k,n) (which we call bridge polytopes) which provide a positive Grassmannian analogue of the permutohedron. In this setting, BCFW bridge decompositions of reduced plabic graphs play the role of reduced decompositions. We define Br(k,n) and explain how paths along its edges encode BCFW bridge decompositions of the longest element pi(k,n) in the circular Bruhat order. We also show that two-dimensional faces of Br(k,n) correspond to certain local moves for plabic graphs, which by a result of Postnikov [Pos06], connect any two reduced plabic graphs associated to pi(k,n). All of these results can be generalized to the positive parts of Schubert cells. A useful tool in our proofs is the fact that our polytopes are isomorphic to certain Bruhat interval polytopes. Conversely, our results on bridge polytopes allow us to deduce some corollaries about the structure of Bruhat interval polytopes.