Shelling totally nonnegative flag varieties

By Lauren K. Williams. 2005

63 citations recorded by the source corpus.

Abstract Lusztig has recently extended the theory of total positivity by introducing the totally nonnegative part (G/PJ )≧0 of an arbitrary flag variety G/PJ . In this paper we study the face partially ordered set (poset) 𝒬 J of cells in Rietsch's cell decomposition of (G/PJ )≧0. Our goal is to use combinatorial techniques to understand what (G/PJ )≧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/PJ )≧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.

Subfields: Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/210e558e072593ebb38cd18decaab4b52d23309f

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Citations in the corpus

Poset Topology: Tools and Applications (2006, methodology): “Dyer’s linear order (called reflection order) was recently used by Williams [210] to obtain an EL-labeling of the poset of cells in a certain cell decomposition of Rietsch [144] of the totally nonnegative part of an arbitrary flag variety (for any reductive linear algebraic group).”
Regularity theorem for totally nonnegative flag varieties (2019, background): “This includes the shellability of Q due to Williams [40], and some topological results [31, 21] on Richardson varieties.”
Scattering Amplitudes and the Positive Grassmannian (2012, background): “We will not prove that ∂ defines an Eulerian poset (this was proven in [104]), but let us at least demonstrate that ∂(2) = 0 (mod 2)—which is of course a prerequisite for ∂ to actually have the meaning of a homological ‘boundary’ operator.”
Cluster algebras (2014, unknown): “In fact, it is a regular CW decomposition [51, 69, 54, 22].”
Total positivity for cominuscule Grassmannians (2007, background): “But in fact it has been shown that the ⊕-diagrams satisfying the (type A) Γ -condition are in bijection with pairs (x,w) where x ∈ W , w ∈ W J , and x ≤ w [12, 22].”