A positive Grassmannian analogue of the permutohedron

Abstract:

The classical permutohedron $\operatorname {Perm}_n$ is the convex hull of the points $(w(1),\dots ,w(n))\in \mathbb {R}^n$ where $w$ ranges over all permutations in the symmetric group $S_n$. 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 $\operatorname {Perm}_n$ is the Hasse diagram of the weak Bruhat order. Equivalently, the paths from $e$ to $w_0$ along the edges of $\operatorname {Perm}_n$ 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 $\operatorname {Br}_{k,n}$ (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 $\operatorname {Br}_{k,n}$ 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 $\operatorname {Br}_{k,n}$ correspond to certain local moves for plabic graphs, which by a result of Postnikov, 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.