Shelling Totally Nonnegative Flag Varieties

In this paper we study the partially ordered set Q of cells in Rietsch’s [20] cell decomposition of the totally nonnegative part of an arbitrary flag variety P ≥0 . Our goal is to understand the geometry of P ≥0 : Lusztig [13] has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P ≥0 is homeomorphic to a ball. The order complex ‖Q‖ is a simplicial complex which can be thought of as a combinatorial approximation of P ≥0 . Using combinatorial tools such as Bjorner’s EL-labellings [1] and Dyer’s reflection orders [7], we prove that Q is graded, thin and EL-shellable. As a corollary, we deduce that Q is Eulerian and that the Euler characteristic of the closure of each cell is 1. Additionally, our results imply that ‖Q‖ is homeomorphic to a ball, and moreover, that Q is the face poset of some regular CW complex homeomorphic to a ball.