Shelling totally nonnegative flag varieties

Abstract

Lusztig has recently extended the theory of total positivity by introducing the totally nonnegative part (G/P_J)_≧0 of an arbitrary flag variety G/P_J. In this paper we study the face partially ordered set (poset) 𝒬^J of cells in Rietsch's cell decomposition of (G/P_J)_≧0. Our goal is to use combinatorial techniques to understand what (G/P_J)_≧0 and its cell decomposition “look like.” The order complex ∥𝒬^J∥ is a simplicial complex which can be thought of as a combinatorial approximation of (G/P_J)_≧0. Using tools such as Bjorner's EL-labellings and Dyer's reflection orders, we prove that 𝒬^J has the most favorable combinatorial properties: namely, it is graded, thin, and EL-shellable. It follows that 𝒬^J is Eulerian. Additionally, our results imply that ∥𝒬^J∥ is homeomorphic to a ball, and moreover, that 𝒬^J is the face poset of a regular CW complex homeomorphic to a ball. In particular, this paper resolves Postnikov's conjecture that the face poset of the totally nonnegative part of the Grassmannian is shellable and Eulerian.