For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (set-theoretical) cross-section φ : G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w 0 in the Weyl group W. It assigns to any gB a representative g ∈ G together with a factorization into simple root subgroups and simple reflections. The cross-section φ is continuous along the components of Deodhar's decomposition of G/B. We introduce a generalization of the Chamber Ansatz and give formulas for the factors of g = φ(gB). These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety (G/B) ≥0 defined by Lusztig, giving a new proof of Lusztig's conjectured cell decomposition of (G/B) ≥0. We also give minimal sets of inequalities describing these cells.

In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian X = Gr n−k (C n ), as well as the mirror dual Landau-Ginzburg model (X ○ , W ∶X ○ → C), whereX ○ is the complement of a particular anti-canonical divisor in a Langlands dual GrassmannianX = Gr k ((C n ) * ), and the superpotential W has a simple expression in terms of Plücker coordinates [MR13]. Grassmannians simultaneously have the structure of an A-cluster variety and an X -cluster variety [Sco06, Pos]; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps [FZ02, FG06]. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network orare the open positroid varieties in X andX, respectively. To each X -cluster chart Φ G and ample 'boundary divisor' D in X ∖ X ○ , we associate a Newton-Okounkov body ∆ G (D) in R k(n−k) , which is defined as the convex hull of rational points; these points are obtained from the multi-degrees of leading terms of the Laurent polynomials Φ * G (f ) for f on X with poles bounded by some multiple of D. On the other hand using the A-cluster chart Φ ∨ G on the mirror side, we obtain a set of rational polytopes -described in terms of inequalities -by writing the superpotential W as a Laurent polynomial in the A-cluster coordinates, and then "tropicalising". Our first main result is that the Newton-Okounkov bodies ∆ G (D) and the polytopes obtained by tropicalisation on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of) these Newton-Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton-Okounkov bodies, in the case that the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians [FW04].

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We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in G L n GL_n forms a real semi-algebraic cell of dimension n − 1 n-1 . Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of G L n ( C ) GL_n(\mathbb {C}) relying in particular on the positivity of the structure constants, which are enumerative Gromov–Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson’s which we explain with proofs in the type A A case.