Positive configuration space

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang-Mills scattering amplitudes, which will be discussed in a sequel.


Introduction
This is the first in a sequence of papers where we define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. In this paper, we focus on the combinatorics and topology of this space. In a sequel [ALS+], we will further study the geometry and its relations to cluster algebras, canonical bases, and scattering amplitudes. Some of the applications of our work to N = 4 super Yang-Mills amplitudes were announced in the note [ALS].
(2) The set D(k, n) of regular subdivisions of the hypersimplex ∆(k, n) into positroid polytopes. (3) The set of cones in the positive tropical Grassmannian Trop >0 Gr(k, n), the space of valuations of positive Puiseux series points Gr(k, n)(R >0 ). (4) The set of cones in the space Dr(k, n) >0 ⊂ R ( [n] k ) (called the positive Dressian) of vectors satisfying the positive tropical (three-term) Plücker relations.
Our second main result is a description of the topology of Ch(k, n) ≥0 , a variant of the results of [Pos, GKLa, GKLb]. Somewhat surprisingly, while the geometry of the Chow quotient is considerably more complicated than that of the Grassmannian, the following result is easier than its Grassmannian counterpart.
There is a stratification-preserving homeomorphism from nonnegative configuration space to a polytope. In particular, each positive Chow cell Θ∆ ,>0 ⊂ Ch(k, n) ≥0 is homeomorphic to an open ball.
We remark that Gr(k, n) ≥0 is not homeomorphic to a polytope as a stratified space. We now explain each of the objects in Theorem 1.1 in turn. Let [n] := {1, 2, . . . , n} and let [n] k denote the set of all k-element subsets of [n]. 1.4. Each point X ∈ Ch(k, n) is represented by an algebraic cycle inside the Grassmannian. If X ∈C onf(k, n) ⊂ Ch(k, n) then X is represented by the torus orbit closure T · V of a generic point V in the Grassmannian. The toric variety T · V is isomorphic to the projective toric variety X ∆(k,n) associated to the hypersimplex ∆(k, n), the convex hull of all vectors e I ∈ R n , I ∈ [n] k with k 0-s and (n − k) 1-s. A general point X ∈ Ch(k, n) is represented by a union of toric varieties X P 1 , . . . , X Pm , where P 1 , . . . , P m are polytopes that form a regular subdivision of the hypersimplex into matroid polytopes (see §2). We thus obtain a stratification of Ch(k, n) by matroid subdivisions of the hypersimplex, see [Kap, KT, Laf]. However, it is a difficult question to describe which matroid subdivisions of the hypersimplex occur in this way. This is a variant of the (also difficult) question of which matroids are realizable.
When X ∈ Ch(k, n) ≥0 is nonnegative, the matroid polytopes P 1 , . . . , P m are positroid polytopes (Proposition 6.1). Since positroids have been completely classified, subdivisions of the hypersimplex by positroid polytopes are far more tractable. Indeed, Theorem 1.1 states that any regular subdivision of the hypersimplex ∆(k, n) into positroid polytopes appears in nonnegative configuration space. 1.5. By definition, a regular subdivision of the hypersimplex arises from a weight vector p • ∈ R ( [n] k ) : we lift each vertex e I of ∆(k, n) to height p I , and project the lower faces of the resulting convex hull down to obtain a subdivision∆(p • ). Speyer [Spe] showed that the subdivision∆(p • ) is into matroid polytopes if and only if p • satisfies the three-term tropical Plücker relations.
We say that p • satisfies the three-term positive tropical Plücker relations if for every S and a < b < c < d not contained in S, we have that (1.1) p Sac + p Sbd = min(p Sab + p Scd , p Sad + p Sbc ).
We show in Proposition 8.3 that the subdivision∆(p • ) is into positroid polytopes if and only if p • satisfies (1.1). Following [HJJS, HJS, OlPS], we call the space of vectors satisfying (1.1) the positive Dressian, and denote it by Dr(k, n) >0 . More generally, for each positroid M, we have a positive local Dressian Dr(M) >0 .
1.6. Speyer and Sturmfels [SS] studied the tropical Grassmannian Trop Gr(k, n) which parametrizes tropical linear spaces. Every point p • ∈ Trop Gr(k, n) satisfies the three-term tropical Plücker relations, but in general the converse is not true [HJJS]. Speyer and Williams [SW] defined the positive tropical Grassmannian Trop >0 Gr(k, n) ⊂ Trop Gr(k, n). Let R = ∞ n=1 R((t 1/n )) denote the field of Puiseux series and R >0 ⊂ R those Puiseux series whose leading (lowest) coefficient is positive. Then Trop >0 Gr(k, n) is defined to be the closure of the set of valuations p • = (p I = val(∆ I (V )) | I ∈ [n] k ) for V ∈ Gr(k, n)(R >0 ). We generalize this by also considering the positive tropical positroid cell Trop >0 Π M . It is immediate that every point p • ∈ Trop >0 Gr(k, n) or p • ∈ Trop >0 Π M satisfies the three-term positive tropical Plücker relations (1.1). A key technical result is that the converse holds (Theorem 9.2).
We give two proofs of Theorem 1.3. The first one uses (1.1) directly. The second one uses a tropical bridge reduction (Section §14) for positive tropical Plücker vectors, a variant of the bridge reduction algorithm for points V ∈ Gr(k, n) ≥0 in [Lam]. Whereas usual bridge reduction gives parametrizations of Π M,>0 , tropical bridge reduction gives parametrizations of Dr(M) >0 = Trop >0 Π M .
1.8. Finally, let us explain some ingredients of the proof of Theorem 1.2. We use the notion of nearly convergent functions on Ch(k, n) (the nomenclature comes from the stringy integrals of [AHLa]). These are certain T -invariant, subtraction-free, rational functions on the Grassmannian whose tropicalizations take nonnegative values. The ring C[Γ] generated by nearly convergent functions is isomorphic to the coordinate ring of an affine open subset X ′ P (k,n) of a projective toric variety X P (k,n) (Proposition 11.8) associated to some polytope P (k, n). We obtain a morphism ϕ : Ch(k, n) −→ X ′ P (k,n) from an open subset Ch(k, n) ⊂ Ch(k, n) of the Chow quotient to X ′ P (k,n) . We show that the restriction of ϕ to the nonnegative part Ch(k, n) ≥0 is a homeomorphism onto the nonnegative part X P (k,n),≥0 of the toric variety, which is known to be homeomorphic to the polytope P (k, n). Organization. In §2, we discuss matroids, positroids, and their matroid polytopes. In §3, we discuss the Grassmannian, configuration space, and the positroid stratification. In §4, we review cluster parametrizations of positroid cells. In §5 and §6, we introduce our main object of interest: the nonnegative configuration space and its stratification by positive Chow cells. In §7 and §8, we study positive tropical vectors and positroid subdivisions of the hypersimplex. In §9, we show that the positive Dressian and the positive tropical Grassmannian agree. In §10 we show that a number of fan structures on the positive Dressian coincide. In §11, we introduce nearly convergent functions. In §12, we prove that Ch(k, n) ≥0 is homeomorphic to a ball. The case k = 2 is studied as an example in §13. In §14, we introduce and study tropical bridge operations. §15 contains some technical statements concerning connected positroids. Appendix A contains data for the cases (k, n) = (3,6), (3,7), (3,8).
Acknowledgements. We thank Song He for conversations related to this work. We are grateful to Lauren Williams for a correction (see Remark 7.1). T.L. was supported by NSF DMS-1464693, NSF DMS-1953852, and by a von Neumann Fellowship from the Institute for Advanced Study. N.A-H. and M.S. were supported by DOE grants DE-SC0009988 and DE-SC0010010 respectively. Remark 1.4. Many of the results in this work were announced at Amplitudes 2019 [Namp]. Some of the results in §7- §9 regarding the positive tropical Grassmannian, the positive Dressian, and positroidal subdivisions of the hypersimplex are not surprising to experts and overlap with independent recent work in [Ola, Eara, Earb, LPW, SW+]. For instance, Proposition 7.2 is closely related to [LPW,Theorem 3.8] and Proposition 8.3 (2) is [LPW,Theorem 9.12].

A matroid M ⊂ [n]
k of rank k on [n] is a nonempty collection of k-element subsets of [n], called bases, satisfying the exchange axiom: The uniform matroid is the collection M = [n] k of all k-element subsets of [n].
2.2. The matroid polytope P M of a matroid in M is the convex hull of the vectors e I , for I ∈ M. Here, e I = e i 1 + e i 2 + · · ·+ e i k is the sum of k basis vectors, where I = {i 1 , . . . , i k }.
Thus the matroid polytope of the uniform matroid is the hypersimplex ∆(k, n), whose vertices are exactly the 0-1 vectors with k 1-s and (n − k) 0-s. We have the following characterization of matroid polytopes.

Proposition 2.1 ([GGMS]).
A polytope P ⊂ R n is the matroid polytope of a matroid of rank k on [n] if and only if its vertex set is a subset of {e I | I ∈ [n] k }, and all edges of P are in the direction of e i − e j , for i = j.

2.3.
If M 1 is a matroid on a set S 1 and M 2 a matroid on S 2 , then the direct sum M 1 ⊕ M 2 is a matroid on the disjoint union S 1 ⊔ S 2 , given by We say that a matroid M of rank k on [n] is connected if the matroid polytope P M is of full dimension, that is, has dimension n − 1. This is equivalent to the condition that M is not a non-trivial direct sum of smaller matroids.

The Bruhat partial order on [n]
k is defined as follows. For two subsets I, J ∈ [n] k , we write I ≤ J if I = {i 1 < i 2 < · · · < i k }, J = {j 1 < j 2 < · · · < j k } and we have i r ≤ j r for r = 1, 2, . . . , k. For I ∈ [n] k , the Schubert matroid S I is defined as and has minimal element I. For a ∈ [n], let ≤ a denote the cyclically rotated order on [n] with minimum a, which induces a partial order ≤ a on [n] k . Let S I,a := {J ∈ [n] k | I ≤ a J} denote the cyclically rotated Schubert matroid.
2.5. A (k, n)-Grassmann necklace I = (I 1 , I 2 , . . . , I n ) [Pos] is a n-tuple of k-element subsets of [n] satisfying the following condition: for each a ∈ [n], we have (1) with indices taken modulo n. A positroid M is the matroid of a totally nonnegative point in the Grassmannian, and are in bijection with Grassmann necklaces. Oh, Pos]). Let I = (I 1 , I 2 , . . . , I n ) be a (k, n)-Grassmann necklace. Then the intersection of cyclically rotated Schubert matroids (2.2) M I = S I 1 ,1 ∩ S I 2 ,2 ∩ · · · ∩ S In,n is a positroid, and the map I → M I gives a bijection between (k, n)-Grassmann necklaces and positroids of rank k on [n].
2.6. Let M be an arbitrary matroid of rank k on [n] and a ∈ [n]. Then M has a minimum with respect to ≤ a , which is denoted I a (M).
Given a (k, n)-Grassmann necklace I = (I 1 , I 2 , . . . , I n ), we define f I : Z → Z by for a = 1, 2, . . . , n, and extending the domain to Z by setting f (i + n) = f (i) + n for all i ∈ Z.
Proposition 2.4. [KLS] For any (k, n)-Grassmann necklace I, the function f I is a (k, n)bounded affine permutation. The map I → f I gives a bijection between (k, n)-Grassmann necklaces and (k, n)-bounded affine permutations.
Thus we have bijections between positroids of rank k on [n], and (k, n)-Grassmann necklaces, and (k, n)-bounded affine permutations. We write f M := f I(M) .
2.8. A polytope P in {(x 1 , . . . , x n ) | x 1 + x 2 + · · · + x n = k} ⊂ R n is called alcoved [LP] if it is given by the intersection of half spaces of the form is a cyclic interval. The following two results are a special case of the theory of polypositroids [LP+], see also [ARW,Proposition 5.5 and Corollary 5.4]. Proposition 2.5 ([LP+]). Let M be a matroid with matroid polytope P M . Then P M is alcoved if and only if M is a positroid.
Proof. The matroid polytope P S I,a of the rotated Schubert matroid S I is the intersection of the hypersimplex ∆(k, n) with the inequalities for i = 1, 2, . . . , n. By definition, this is an alcoved polytope, and so is the intersection P S I 1 ,1 ∩ P S I 2 ,2 ∩ · · · ∩ P S In,n . Since every positroid is of the form (2.2), every positroid polytope is alcoved. Now let P M be the matroid polytope of an arbitrary matroid. Then the smallest alcoved polytope P containing P M is the intersection of the rotated Schubert matroid polytopes P S Ia(M),a for a = 1, 2, . . . , n. Thus P is the matroid polytope of the positroid envelope of M. In particular, if M is itself a positroid then P M is alcoved. Corollary 2.6. Every face of a positroid polytope is itself a positroid polytope.
A noncrossing partition (S 1 , . . . , S r ) of [n] is a partition of [n] into disjoint sets such that there do not exist a < b < c < d such that a, c ∈ S i and b, d ∈ S j for i = j. Proposition 2.7 ([ARW,Theorem 7.6]). Let f = f M be the bounded affine permutation associated to a positroid M. Suppose that M = M 1 ⊕ M 2 ⊕ · · · ⊕ M r , where M i is a positroid on the ground set S i ⊂ [n]. Then (S 1 , . . . , S r ) form a noncrossing partition of [n], and (f M (S i ) mod n) = S i for i = 1, 2, . . . , r. In particular, the connected components of the positroid M are themselves positroids.
Conversely, let (S 1 , . . . , S r ) be a noncrossing partition of [n] and M i be a positroid on the ground set S i . Then the direct sum M 1 ⊕ M 2 ⊕ · · · ⊕ M r is a positroid.

Grassmannians and configuration spaces
k denote its Plücker coordinates. These Plücker coordinates satisfy the Plücker relations and are defined up to a common scalar. We refer the reader to [Lam] for further details. The most important relation for us will be the three-term Plücker relation is of size k − 2 and a < b < c < d are not contained in S.
It is convenient to also work with the affine cone Gr(k, n) over Gr(k, n). A point V in Gr(k, n) is a collection of Plücker coordinates ∆ I (V ) that satisfy the Plücker relations.
But now scaling Plücker coordinates give different points, and Gr(k, n) also contains a distinguished cone point 0, where all Plücker coordinates vanish.
For the uniform matroid, we have 3.3. The torus (C × ) n = {(t 1 , t 2 , . . . , t n ) | t i ∈ C × } acts on Gr(k, n) by scaling the i-th column of a representing matrix by t i . For t ∈ C × , the element (t, t, . . . , t) ∈ (C × ) n scales all Plücker coordinates by t k , and thus the action of (C × ) n factors through the torus An element (t 1 , . . . , t n ) ∈ (C × ) n acts on V ∈ Gr(k, n) by This action factors through the quotient torus where Z/kZ = {(ζ, . . . , ζ) | ζ k = 1} is a cyclic group of order k. The character lattices X(T ) and X( T ) of T and T are naturally identified with sublattices of Z n = X((C × ) n ): Proof. Suppose M is connected. We may find a vertex e I of P M and vertices e J 1 , . . . , e J n−1 connected to e I via an edge of P M , so that the span of {e I , e J 1 , . . . , e J n−1 } is linearly independent. By Proposition 2.1, the edges of P M are roots, i.e., vectors of the form e i − e j . A collection of linearly independent roots is easily seen to be unimodular, i.e., their integral span is the lattice {(x 1 , . . . , x n ) ∈ Z n | i x i = 0}, which is the character lattice X(T ) of T . Points V ∈ G M can be gauge-fixed to satisfy ∆ I (V ) = 1. Under this gauge-fix, the torus T acts on the Plücker coordinates ∆ J 1 , . . . , ∆ J n−1 with weights e J 1 − e I , . . . , e J n−1 − e I . We have just argued that these weights span the character lattice X(T ), and thus T must act freely on G M .
The "if" direction is similar.
3.4. The torus T acts freely on the complex manifoldGr(k, n), and thus the quotient Gr(k, n)/T is a manifold of dimension k(n − k) − (n − 1). The spaceGr(k, n)/T has the following alternative description. Let Conf(k, n) be the space (P k−1 ) n /GL(n) of GL(n)orbits of n points in the projective space P k−1 . A configuration p = (p 1 , . . . , p n ) is called generic if any r of the points, for r ≤ k, affinely span a subspace of P k−1 of dimension r − 1. We denote the space of generic configurations byC onf(k, n) ⊂ Conf(k, n). We have an isomorphismGr(k, n)/T ≃C onf(k, n).
3.5. The totally nonnegative Grassmannian Gr(k, n) ≥0 is the subspace of Gr(k, n) consisting of points V ∈ Gr(k, n)(R) all of whose Plücker coordinates are nonnegative. The totally positive Grassmannian Gr(k, n) >0 consists of points all of whose Plücker coordinates are positive. These spaces were defined by Lusztig [Lus] and Postnikov [Pos]. The totally nonnegative Grassmannian Gr(k, n) ≥0 is homeomorphic to a closed ball [GKLa]. The matroid of a point V ∈ Gr(k, n) ≥0 is called a positroid. Positroids can be indexed by (k, n)-Grassmann necklaces (Proposition 2.2) and (k, n)-bounded affine permutations (Proposition 2.4). The matroid of V ∈ Gr(k, n) >0 is the uniform matroid, and thus we have Gr(k, n) >0 ⊂Gr(k, n). Indeed, Gr(k, n) >0 is a connected component of the manifold Gr(k, n), and is diffeomorphic to an open ball of dimension k(n − k) − (n − 1). The image of Gr(k, n) >0 inGr(k, n)/T ≃C onf(k, n) is called the positive component of (generic) configuration space, and denoted Conf(k, n) >0 . By [Pos], Π M,>0 is homeomorphic to an open ball. We define dim(M) to be the dimension of this ball. We have the disjoint union [Pos] (3.3) gives Gr(k, n) ≥0 the structure of a regular CW complex [GKLb].
3.7. The group T >0 := R n >0 /R >0 ⊂ (C × ) n /(C × ) = T acts on Gr(k, n) ≥0 , preserving the positroid cells Π M,>0 . From Lemma 3.1, we have the following. 3.9. We recall the bridge parametrizations of Π M from [Lam], see also [Kar]. The group GL(n) acts on Gr(k, n) by right multiplication. For i = 1, 2, . . . , n − 1, let x i (t) ∈ GL(n) be the matrix that differs from the identity in a single entry equal to t in the (i, i + 1)-th matrix entry. For i = n, we let x n (t) ∈ GL(n) be the matrix that differs from the identity in a single entry equal to (−1) k−1 t in the (n, 1)-th matrix entry. Thus x i (t) acts on a k × n matrix V by adding the i-th column to the (i + 1)-th column (with a sign if i = n). The action of x i (t) can be written in Plücker coordinates as (cf. [Lam,Lemma 7.6]) otherwise.
More generally, for γ = (i, j) let x γ (t) be the matrix that differs from the identity in a single entry equal to ±t in the (i, j)-th matrix entry, taking the positive sign if i < j and the sign (−1) k−1 if i > j.
The following result allows us to reduce totally nonnegative points recursively.
Then at least one of the following holds: Then V is in the image of the map κ i : Gr(k, n − 1) ≥0 ֒→ Gr(k, n) ≥0 obtained by adding an i-th column equal to 0.
where V ′ ∈ Π M ′ ,>0 with M ′ the positroid satisfying f M ′ = f M s i (where s i is the simple transposition swapping i and i + 1), and ≃ Π M,>0 obtained in this way are called bridge parametrizations. Note that κ i+1 • x i (t) = x i,i+2 (t) • κ i+1 , so in general we need to use the matrices x γ (t). Bridge parametrizations are of the form where d = dim(M) and x i 1 ,...,i k = span(e i 1 , . . . , e i k ) ∈ Gr(k, n) T is a torus fixed point. We caution that in general x γ (t) for t > 0 does not preserve total nonnegativity. They do preserve total nonnegativity when used in a bridge parametrization. k is the uniform matroid, then we simply call C a cluster. Any cluster C for M has cardinality |C| = dim(M) + 1. (We consider only clusters where the cluster variables are Plücker coordinates coming from face labels of a plabic graph.) Clusters for M can be described via weak separation [OhPS]. We say that two subsets I, J ⊂ [n] are weakly-separated if we cannot find 1 ≤ a < b < c < d ≤ n such that a, c ∈ I \ J and b, d ∈ J \ I (or with I and J swapped). The following result can be taken to be the definition of a cluster. Every I ∈ [n] k belongs to some cluster C ⊂ [n] k , but this is not true with an arbitrary positroid M replacing [n] k . 4.2. Let S ⊂ [n] be of size k − 2 and a < b < c < d numbers not contained in S. Let C ∈ M be a cluster. If Sac, Sab, Scd, Sad, Sbc ∈ C (resp. Sbd, Sab, Scd, Sad, Sbc ∈ C) then we can mutate C at Sac (resp. Sbd) to produce another cluster C ′ ⊂ M where Sac has been replaced by Sbd (resp. Sbd has been replaced by Sac). The Plücker variables of C and C ′ are then related by (3.1).

Proposition 4.2 ([OhPS]
). Any two clusters C, C ′ ⊂ M (as in Proposition 4.1) are related by a sequence of mutations.
By a positive Laurent polynomial we mean a Laurent polynomial such that the coefficient of every monomial is nonnegative. The following result is a special case of general positivity results of cluster algebras [LS].
k , and a cluster C ⊂ [n] k , the Plücker variable ∆ J is a positive Laurent polynomial in {∆ I | I ∈ C}.
For an arbitrary positroid M, we have the following weaker statement. Proof. This result follows, for example, from the formulae in [MS]. It also follows from the proof of Theorem 7.3 we give below. Namely, in that proof we show that a formula for ∆ J in terms of {∆ I | I ∈ C} can be obtained by iteratively applying the three-term Plücker relation (3.1). In other words, we iteratively substitute ∆ Sac = (∆ Sab ∆ Scd + ∆ Sad ∆ Sbc )/∆ Sbd , without ever dividing by 0.
The following conjecture is likely known to many experts. It does not immediately follow from the identification [GL] of C[Π M ] with a cluster algebra because there are Plücker variables ∆ I , I ∈ M that are not cluster variables in any cluster.
is a homeomorphism onto Gr(k, n) >0 , and every Plücker coordinate is a subtraction-free rational expression ∆ I (x) in x 1 , . . . , x m . Any choice of cluster C gives a positive parametrization after setting one of the Plücker coordinates to 1: the map , called the Laurent phenomenon.

We now consider a simple-minded notion of "cluster" for Π
We shall prove the following result in §15.3.
Lemma 4.6. Let M be a connected positroid. Then there exists a cluster C ⊂ M such that a gauge-fix G ⊂ C exists.
If G is a gauge-fix then the action of T can uniquely fix ∆ J = 1 for all J ∈ G (assuming that ∆ J = 0 for all J ∈ G). We thus have a canonical identification

Nonnegative configuration space
The goal of this section is to construct a compactification of Conf(k, n) >0 .
5.1. Let X ⊂ P k−1 be an irreducible subvariety of dimension r −1. The degree deg(X) of X is equal to the number #(L ∩ X) of intersection points of X with a generic hyperplane , to be the unique up to scalar non-zero element of R d (k − r, k) that vanishes on Z(X). The variety X can be recovered from R X .

5.
2. An (r − 1)-dimensional algebraic cycle in P k−1 is a formal finite linear combination X = m i X i , where m i are nonnegative integers and X i ⊂ P k−1 are irreducible closed subvarieties of dimension (r − 1). We define deg(X) = m i deg(X i ). Let C(r, d, k) denote the set of all (r − 1)-dimensional algebraic cycles in P k−1 of degree d. Then C(r, 1, k) can naturally be identified with the Grassmannian Gr(r, k) of r-planes in R k . The set C(r, d, k) acquires the structure of an algebraic variety, called the Chow variety, via the following result of Chow and van der Waerden.
Theorem 5.1. The map C(r, d, k) → P(R d (k − r, k)) given by X → i R m i X i defines an embedding of C(r, d, k) as closed subvariety of P(R d (k − r, k)).

A special case of Theorem 5.1 is the statement that Gr
Since T · V ≃ T , the variety X has dimension (n − 1). It is a toric variety with moment polytope equal to the hypersimplex. It follows that the degree of X inside Gr(k, n) and inside P ( [n] k ) −1 is equal to the volume Vol(∆(k, n)), the Eulerian number A n,k . We thus have a natural injectionC onf(k, n) ֒→ C(n − 1, A n,k , n k ) sending a point V ∈C onf(k, n) to the algebraic cycle T · V . The closure ofC onf(k, n) in C(n−1, A n,k , n k ) is called the Chow quotient of the Grassmannian, and denoted Ch(k, n). It is a projective algebraic variety.
We define the totally nonnegative part of the Chow quotient of the Grassmannian or nonnegative configuration space, denoted Ch(k, n) ≥0 , to be the closure of the image of Conf(k, n) >0 in Ch(k, n). It is a compact Hausdorff topological space.
Remark 5.2. For a positroid M, we can also define Ch(M) ≥0 as the closure of Π M,>0 /T >0 inside an appropriate Chow variety.

Positive Chow cells
A point X ∈ Ch(k, n) is an algebraic cycle in Gr(k, n) of dimension (n − 1) and degree A n,k . We have X = s i=1 m i X i where X i are toric varieties that are torus-orbit closures for the same torus T , and we assume m i are positive. Let P i denote the moment polytope of X i . By [Kap], we have the following constraints on X: (1) we have m i = 1 for all i, (2) each P i = P M i is a matroid polytope, and (3) the polytopes P 1 , P 2 , . . . , P s form a regular polyhedral subdivision of the hypersimplex ∆(k, n). For X ∈ Ch(k, n) ≥0 , we strengthen this result as follows.
. Then X i is a toric variety with moment polytope equal to a positroid polytope P i = P M i . The positroid polytopes P 1 , . . . , P s form a regular polyhedral subdivision of the hypersimplex ∆(k, n).
. . which all belong to Gr(k, n) >0 . It follows that q ∈ Gr(k, n) ≥0 , and thus the matroid of X i is a positroid.
Let D(k, n) denote the set of regular polyhedral subdivisions of the hypersimplex into positroid polytopes. For X ∈ Ch(k, n) ≥0 , let∆(X) denote the positroid decomposition from Proposition 6.1. We have a stratification Define Θ∆ ,≥0 to be closure Θ∆ ,>0 in the analytic topology. The spaces Θ∆ ,>0 and Θ∆ ,≥0 are analogues of the open and closed positroid cells Π M,>0 and Π M,≥0 respectively. Define a partial order on D(k, n) by∆ ′ ≤∆ if∆ ′ is a refinement of∆.
The proof of Theorem 6.2 is delayed to §12.2. Theorem 6.2 generalizes various results concerning the topology of positroid cells [Pos, GKLa, GKLb].

Positive tropical Plücker vectors
k } be a collection of "numbers" where p I ∈ R ∪ {∞}, and not all p I are equal to infinity. The support of p • is the collection Supp(p) = {I | p I < ∞} ⊂ [n] k . We say that p • satisfies the tropical Plücker relations [Spe] if for every S of size k − 2 and a < b < c < d not contained in S, the minimum of the three quantities {p Sac + p Sbd , p Sab + p Scd , p Sad + p Sbc } is attained twice. We say that p • is a tropical Plücker vector if it satisfies the tropical Plücker relations and, in addition, the support of p • is a matroid. We say that p • satisfies the positive tropical Plücker relations if for every S and a < b < c < d not contained in S, the equation (1.1) Remark 7.1. In an earlier version of this work, we erroneously asserted that any vector p • satisfying the tropical Plücker relations had support Supp(p • ) given by a matroid. We thank Lauren Williams for the correction. See [OlPS] for some related discussion. 7.2. Tropicalization takes a subtraction-free rational expression to a piecewise-linear expression under the substitution For example, the rational function x 3 + y + 1 2xy + y 2 tropicalizes to the piecewise-linear function min(3X, Y ) − min(X + Y, 2Y ). The equation (1.1) is obtained from (3.1) by applying (7.1), and sending the variables ∆ I to the variables p I . 7.3. For a positroid M, let Dr(M) >0 denote the set of positive tropical Plücker vectors with support M. Let Dr(M) >0 (Z) (resp. Dr(M) >0 (Q)) denote those vectors that are integral (resp. rational). We write Dr(k, n) >0 when M is the uniform matroid and call p • ∈ Dr(k, n) >0 a finite positive tropical Plücker vector. We let Dr(k, n) ≥0 denote the set of all positive tropical Plücker vectors, the union of Dr(M) >0 over all positroids M (see Proposition 7.2).
In [HJJS], the set of finite tropical Plücker vectors is called the Dressian, and following their terminology, we call Dr(k, n) >0 the positive Dressian.
7.4. If p • satisfies is a positive tropical Plücker vector, then in addition to (2.1), M = Supp(p • ) satisfies the following positive 3-term exchange relation: Proof. We establish this result by induction on n. Suppose that M is disconnected, so M = M 1 ⊕ M 2 on disjoint ground sets S 1 and S 2 , such that S 1 ∪ S 2 = [n]. If S 1 , S 2 are cyclic intervals, then M i satisfies positive 3-term exchange relation within S i . Thus by induction M is the direct sum of two positroids on disjoint cyclic intervals, and is thus a positroid by Proposition 2.7. Otherwise, we can find direct summands M ′ , M ′′ of M which are two connected matroids on subsets A ⊔ C, B ⊔ D that are crossing, i.e. A, B, C, D are cyclic intervals occurring in cyclic order. Since M ′ is connected, there are bases I 1 , I 2 of M ′ such that |I 1 ∩ A| = |I 2 ∩ A|. By repeated application of the basis exchange axiom, we see that M ′ contains two bases I ′ , J ′ such that J ′ = I ′ ∪ {c} − {a} with a ∈ A and c ∈ C. Similarly, we have bases I ′′ , J ′′ of M ′′ such that J ′′ = I ′′ ∪ {d} − {b} with b ∈ A and d ∈ C. We can thus find bases T ab, T cd of M, while T ac, T bd are not bases, a contradiction.
We now assume that M is a connected matroid. If it is not a positroid, then by Lemma 2.5, its matroid polytope P M has a facet cut out by an equation \ S| ≥ 2 implies that both M 1 and M 2 are non-trivial. Since the ground sets S and [n] \ S are crossing, our earlier argument implies that M ′ cannot satisfy the positive 3-term exchange relation. After a cyclic relabeling we may assume that T ab, T cd ∈ M ′ , while at least one of T ac and T bd is not in M ′ . However, both T ac, T bd are in M, so this is only possible if both e T ac and e T bd do not lie on the hyperplane H. Indeed, the two vertices e T ac and e T bd lie on opposite sides of H, and this contradicts the assumption that H is a facet. 7.5. Whereas the space of tropical Plücker vectors has a very complicated polyhedral structure [SS, HJJS], the situation is much simpler for positive tropical Plücker vectors.
Theorem 7.3. Let M be a positroid and C be a cluster for M. The maps We suppose by induction that the result has been proven for all there is a unique noncrossing matching on these 2d points, which after reindexing we assume to be {(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i d , j d )} where i r < a j r for all r. Here, < a denotes the cyclic rotation of the total order where a is minimal. Since (I a , J) is not weakly separated, we can find (i, j) and (i ′ , j ′ ) in this matching so that i < a j < a i ′ < a j ′ and there are no elements of (I a \ J) ∪ (J \ I a ) in the open cyclic intervals (i, j) and (i ′ , j ′ ). Let so that we have a positive tropical Plücker relation We make the following claims: for each t = 0, 1, 2, 3, 4, one of the following holds: In particular, for L = I a , we have and for L = J, we have for some c, We say that (x, y) ∈ [n] 2 crosses (u, v) ∈ [n] 2 if all of x, y, u, v are distinct and the two line segments xy and uv cross when 1, 2, . . . , n are arranged in order around a circle. Suppose that (I b , K t ) is not weakly separated. Then we have x, y ∈ I b \K t and u, v . Then i, i ′ ∈ I b so neither u or v is equal to i, i ′ , and so the claim follows from (I b , J) being weakly separated. Case since they would have to belong to I a and thus also to J.
. We may assume that u = i and since i ′ ∈ I b , we have v ∈ J \ I b . Neither x nor y lies in (i, j) since they would have to belong to I a and thus also to J. But j ∈ J \ I a and by (7 Thus (x, y) crosses something of the form (j, r) or (j ′ , r) where r ∈ J \ I b . This contradicts (I b , J) being weakly separated. Proof of (2). We use the description of M as an intersection of Schubert matroids (Proposition 2.2). We prove that K 3 , K 4 ∈ M (and this immediately implies K 0 ∈ M). Indeed, we show that swapping any i < a j where the open interval (i, j) contains nothing in (I a \J)∪(J \I a ) works. So let J ′ be the result of such a swap. We need to show that J ′ ≥ b I b for all b, and it suffices to show this for b ∈ J ′ . Let L = (i, j)∩I a = (i, j)∩J = (i, j)∩I a ∩J.
Case (a): Suppose b ≤ a i. The claim follows from J ≥ b I b together with {i} ∪ L ⊂ I b , which holds since {i} ∪ L ⊂ I a , using (7.5). Case Proof of (3). Suppose that K t ∈ M. Then from (1) we have w(K t ) ≤ w(J), and equality can only happen if (I a , K t ) is not weakly separated. But by construction we have We have proved all the claims (1),(2), (3). By induction, all of the p Kt are determined by p I , I ∈ C. The formula p J = min(p K 1 + p K 2 , p K 3 + p K 4 ) − p K 0 shows that p J is also determined by p I , I ∈ C. By induction, p • ∈ Dr(M) >0 is determined by p I , I ∈ C. Furthermore, it is clear that if p I ∈ Z for I ∈ C then p • ∈ Dr(M) >0 (Z). The theorem is proven.
Another proof of Theorem 7.3 is given after Theorem 9.3. Let X ∨ ( T ) ⊂ R n denote the lattice generated by Z n and the vector (1/k, 1/k, . . . , 1/k) ∈ R n . The lattice X ∨ ( T ) is dual to the lattice X( T ) ⊂ Z n of (3.2). Restricting the action Corollary 7.4. Let M be a connected positroid, C be a cluster for M and G ⊂ C be a gauge-fix. The maps We prove the Z case. The statement follows from Theorem 7.3 and the following claim: given p • ∈ Dr(M) >0 (Z) and integers (c J | J ∈ G) ∈ Z G , there is a unique p ′ • ∼ p • such that p ′ J = c J for all J ∈ G. This claim follows from the following statement: for any (c J | J ∈ G) ∈ Z G , there exists a ∈ X ∨ ( T ) such that a · e J = c J for all J ∈ G, which in turn follows from the definition of gauge-fix in §4.4.

Subdivisions of the hypersimplex
the union of all Q ∈P is equal to P . We typically give a subdivision by only listing the polytopes Q of maximal dimension. 8.2. Given any p • ∈ R ( n k ) we obtain a subdivision of the hypersimplex as follows. We lift each vertex e I ∈ ∆(k, n) to the point e ′ I = (e I , p I ) in one higher dimension. Then we project the lower faces of the convex hull Conv(e ′ I ) back into ∆(k, n). These faces will give us a polyhedral subdivision of ∆(k, n) denoted∆(p • ), and these subdivisions are called regular. More generally, for p • ∈ (R ∪ {∞}) ( n k ) (not all equal to ∞), we obtain a regular subdivision∆(p • ) of the polytope P Supp(p•) := Conv(e I | I ∈ Supp(p • )). By lifting some vertices e I to ∞, the resulting convex hull Conf(e ′ I ) becomes a polyhedron with many "vertical" faces. The projection of the lower faces of Conv(e ′ I ) will only cover P Supp(p•) . Let us describe the faces of∆(p • ) more explicitly. Faces F of∆(p • ) are convex polytopes whose vertices are a subset of the vectors {e I | I ∈ [n] k }. Abusing notation, we will also consider F as a subset of [n] k . We call p • and p ′ • equivalent, and write p • ∼ p ′ • if there exists a vector a = (a 1 , . . . , a n ) ∈ R n such that for all I ∈ [n] k , (8.1) Proof. If F is full-dimensional, then the vectors {e I | I ∈ F } span R n . Thus at most one p ′ • equivalent to p • satisfies the condition p ′ I = 0 for I ∈ F . The existence and the last conclusion follows from the assumption that F is a face of∆(p • ).
The following result is immediate.
Lemma 8.2. The condition that p • is a tropical Plücker vector (resp. positive tropical Plücker vector) is a property of the equivalence class of p • .
8.3. Suppose that p • is a tropical Plücker vector. Then∆(p • ) is a regular subdivision of the matroid polytope P Supp(p•) . Suppose that p • is a positive tropical Plücker vector. Then by Proposition 7.2,∆(p • ) is a regular subdivision of the positroid polytope P Supp(p•) . Part (1) of the following result is proved in [Spe]; see also [OlPS,Corollary 13]. Proof. We prove (2). Suppose that∆(p • ) is a positroid subdivision. Let us fix S ⊂ [n] of size k − 2 and a < b < c < d not contained in S. The intersection of ∆(k, n) with the hyperplanes {x i = 0 | i / ∈ Sabcd} and {x i = 1 | i ∈ S} is a face F = F (S; abcd) of ∆(k, n). This face F is an octahedron with six vertices e Sab , e Sac , e Sad , e Sbc , e Sbd , e Scd . The intersection of the positroid subdivision∆(p • ) with F gives a subdivision ofF (or a subpolytope of F ), which must be a positroid subdivision. Thus it suffices to observe that the positive tropical Plücker relation (1.1) holds for the six "numbers" p Sab , p Sac , p Sad , p Sbc , p Sbd , p Scd if they induce a positroid subdivision of a subpositroid ofF . This is a straightforward case-by-case analysis.
Suppose that p • satisfies the positive tropical Plücker relation. Let F be a face of the subdivision∆(p • ). Then we can find p ′ (1), we know that F = Supp(q • ) is a matroid. By Proposition 7.2, to show that F is a positroid, it suffices to show that q • is a positive tropical Plücker vector. Consider any S and a < b < c < d as in (1.1). If both sides of (1.1) are equal to 0 (resp. positive) for p ′ • , then both sides of (1.1) are equal to 0 (resp. equal to ∞) for q • . Thus (1.1) holds for q • if it holds for p ′ • .
9. Positive tropical Grassmannian 9.1. Let R = ∞ n=1 R((t 1/n )) denote the field of Puiseux series over R. We define val : R → R ∪ {∞} by val(0) = ∞ and val(x(t)) = r if the lowest term of x(t) is equal to αt r where α ∈ R × . We define R >0 ⊂ R to be the semifield consisting of Puiseux series x(t) that are non-zero and such that coefficient of the lowest term is a positive real number. We let R ≥0 := R >0 ∪ {0}.
Proof. The support of p • is a matroid M. The three-term positive tropical Plücker relations follows from tropicalizing (3.1). Thus p • is a positive tropical Plücker vector.
For a positroid M, we also define By Proposition 7.2 and Lemma 9.1, we have the disjoint union as M varies over positroids of rank k on [n], an analogue of (3.3).
9.2. We now connect Gr(k, n)(R >0 ) to the nonnegative part Ch(k, n) ≥0 of the Chow quotient. Let γ(t) be a curve in Gr(k, n) that is analytic near t = 0. Thus each Plücker coordinate ∆ I (γ(t)) has a Taylor expansion g I (t) ∈ R[[t]] that converges near t = 0. Suppose that for some s > 0, we have that γ((0, s)) ∈ Gr(k, n) >0 . Then for every I, we have that g I (t) = 0 and the coefficient of the lowest term of g I (t) must be positive. In particular, γ(t) "agrees" with a point V (t) ∈ Gr(k, n)(R >0 ). Furthermore, γ((0, s)) determines a curve in Conf(k, n) >0 = Gr(k, n) >0 /T >0 , and thus lim t→0 γ(t) is a point in X = i X i ∈ Ch(k, n) ≥0 . The hypersimplex decompositioñ ∆(X) is given by∆(X) =∆(val(V (t))). Let d(a) = diag(t a 1 , . . . , t an ) ∈ GL(n)(R) be an R-valued point in the torus acting on Gr(k, n). Then p • = val(V ) and p ′ • = val(V · d(a)) are related by (8.1). Let V ′ (t) = V (t) · d(a), and suppose that p ′ • = val(V ′ (t)) ≥ 0 and at least one p I is equal to 0. Then V 0 = lim t→0 V ′ (t) lies in Π M,>0 for some positroid M. The point V 0 belongs to (at least) one of the toric varieties X i = T · V i , and the positroid polytope P M is one of the faces of∆(X). To obtain the maximal faces, one has to pick the vector a carefully, just as (8.2) typically gives lower-dimensional faces.
We refer the reader to [KT] for further details from this perspective.
surjectively onto Q d . By Theorem 7.3 we have ν C : Dr(M) >0 (Q) → Q d is an isomorphism, we see that every rational positive tropical Plücker vector is realizable. Taking the closure in R d , we obtain Trop >0 Π M = Dr(M) >0 . 9.5. We give another proof of Theorem 9.2 using bridge decompositions, which we believe is of independent interest. Recall from (3.5) the notion of bridge parametrizations of Π M,>0 . In §14, we define tropical bridges T i (a), a ∈ Z (and more generally T γ (t) for γ = (i, j)) acting on the space of positive tropical Plücker vectors, with the following property: for a(t) ∈ R >0 and V (t) ∈ Gr(k, n)(R ≥0 ) val(V (t) · x i (a(t))) = T i (val(a(t))) · val(V (t)).
In particular, if p • is representable, it follows immediately that T i (a) · p • is representable for any a ∈ Q. The proof of Theorem 9.3 is given in §14. Theorem 9.2 follows immediately from Theorem 9.3. We give another proof of Theorem 7.3.
Second proof of Theorem 7.3. Any two bridge parametrizations of Π M,>0 are related by an invertible rational transformation that is subtraction-free in both directions, see [Pos]. For particular choices of bridge parametrizations and cluster parametrizations, [MS] gives explicit invertible subtraction-free rational expressions between them; for one direction we have [MS,Theorem 3.3] and for the other combine [MS,Theorem 7.1] with [MS,Proposition 7.10]. Tropicalizing these subtraction-free rational transformations (using (7.1)), we conclude that z ∈ R d and (p I | I ∈ C) ∈ R C are related by an invertible piecewise linear transformation, and this holds even integrally. Thus the bijections of (7.3) follow from Theorem 9.3.

Fan structure
10.1. A (polyhedral) fan F = {C} in a vector space R d is a finite collection of closed polyhedral cones C ⊂ R d satisfying the conditions: (1) If C, C ′ ∈ F then C ∩ C ′ ∈ F .
(2) For C ∈ F , every face C ′ ⊂ C is in F . We say that F is complete if the union of the cones in F is equal to R d . In the case that F is complete, or if F is pure of some dimension d ′ (i.e., all maximal cones have dimension d ′ ), we call the maximal cones chambers. >0 and Dr(k, n) >0 have a number of different fan structures that we now define. All of these fan structures are compatible with equivalence: if p • ∼ p ′ • then they belong to the same cone. Thus we can also think of these as complete fan structures on the vector spaces Dr(M) >0 /∼ and Dr(k, n) >0 /∼. We define an integer dim(∆) for a regular polyhedral subdivision∆ of P M into positroid polytopes:

For every S ⊂ [n]
and a < b < c < d not contained in S, we have the three term Plücker fan whose two chambers are given by This fan has exactly three cones: the two chambers above and a third cone where p Sac + p Sbd = p Sad + p Sbc = p Sab + p Scd , the intersection of the two chambers. The Plücker fan structure on Dr(M) >0 is the common refinement of all the three term Plücker fans. In other words, two vectors p • and q • belong to the same relatively open cone of the Plücker fan structure if exactly which of the three quantities p Sac + p Sbd , p Sad + p Sbc , p Sab + p Scd is minimal is the same for p • and for q • .
10.5. The positive fan structure is the fan whose cones are the images of the domains of linearity for a positive parametrization by a cluster. To be precise, pick a cluster C for M. By Theorem 7.3 we identify p • ∈ Dr(M) >0 with a point in R C . For each J ∈ M, the function p J is a piecewise-linear function on R C . The common domains of linearity (see §11.1 for further discussion) for all the functions p J , J ∈ M is the positive fan structure of Dr(M) >0 with respect to the cluster C. Proof. In each chamber A of the positive fan structure on R C , all the functions p J , J ∈ M are linear. The composition and inverse of linear functions is linear, so under the piecewiselinear isomorphism R C → R C ′ the cone A will be sent inside some chamber A ′ of the positive fan structure on R C ′ , and reversing the roles of C and C ′ we see that A and A ′ are isomorphic.
We may thus speak of the positive fan structure on Dr(M) >0 .
10.6. We shall show that the secondary fan structure, the Plücker fan structure, and the positive fan structure coincide. The secondary fan structure and the Plücker fan structure are shown to coincide in [OlPS]. Proof. We show that the Plücker fan and the positive fan agree. By Proposition 10.2, it suffices to compare cones of maximal dimension dim(Π M ). Let A ⊂ R C be such a cone in the positive fan structure for a cluster C for M. Then all the functions p J are linear functions on A: we have p J = I α A J,I p I , where α A J,I ∈ Z. Substituting these expressions into (p Sac + p Sbd ) − (p Sab + p Scd ) or (p Sac + p Sbd ) − (p Sad + p Sbc ) we obtain two linear functions. For each point of A, at least one of these linear functions vanishes. Since A has maximal dimension, one of the two linear functions is identically 0 on A; it follows that A is completely contained in some chamber of the Plücker fan. We have shown that the positive fan structure is a refinement of the Plücker fan structure.
Next let A ′ be a chamber of the Plücker fan structure. The proof of Theorem 7.3 shows that each p J can be written as a piecewise-linear function of p I , I ∈ C, by iteratively applying the equality (1.1). Within the chamber A ′ , the RHS of (1.1) is a linear function instead of a piecewise-linear function. It follows that the restriction of p J to A ′ is a linear function. We have shown that the Plücker fan structure is a refinement of the positive fan structure.
Remark 10.4. In [SW] a particular fan structure on Trop >0 Gr(k, n) is studied. The positive parametrization in [SW] is related by an invertible monomial transformation to a gauge-fix of the cluster parametrization for the cluster The tropicalization of a monomial transformation is a linear map, so the fan structure of [SW] agrees with the one in Theorem 10.3. 10.7. Theorem 10.3 allows us to parametrize D(k, n) with collections of planar trees, see also [HJJS,Section 4] and [BC, CGUZ]. For a subset S ⊂ [n], let H S=1 denote the subspace given by intersecting {x s = 1} for s ∈ S. Note that the face H S=1 ∩ ∆(k, n) is isomorphic to ∆(k − |S|, n − |S|).
Let∆ Proof. Suppose∆ =∆(p • ) where p • ∈ Dr(k, n) >0 . To determine∆, by Theorem 10.3 we need to know which cone of the Plücker fan structure p • lies in. Thus for every S ⊂ [n] and a < b < c < d not contained in S, we need to know which of the following situations we are in: Again, by Theorem 10.3, this is determined by∆ S ∈ D(2, [n] \ S) which in turn is determined by T S .

Note however that not every collection {T [n]\S | S ∈ [n]
k−2 } of planar trees are in the image of (10.2).

Nearly convergent functions
For more details on the material of this section we refer the reader to [AHLa]. 11.1. Let f (x) = p(x)/q(x) ∈ R(x) := R(x 1 , . . . , x r ) be a subtraction-free rational function i.e., both p(x) and q(x) are polynomials with positive coefficients. The piecewiselinear function Trop(f ) on R r is obtained by the substitution (7.1).
Suppose now that f (x) ∈ R ≥0 [x ±1 1 , . . . , x ±1 r ] is a positive Laurent polynomial. Let N[f (x)] denote the Newton polytope of f (x) inside R r . This is the lattice polytope given by the convex hull of v ∈ Z r , as v varies over lattice points such that αx v is a monomial appearing in f (x). The following result is well-known.
Thus Trop(f (x)) is nonnegative if and only if Trop(x v /q(x)) is nonnegative for all v such that a monomial α v x v occurs in p(x). Thus it suffices to establish the claim for the case f (x) = x v /q(x), i.e., Trop(f (x)) is nonnegative if and only if v ∈ N[q(x)]. This follows from the following observation: a point v ∈ Z r is outside a polytope P if and only if there exists a vector a ∈ R r such that a · v < min(a · u | u a vertex of P ). 11.3. Set d = k(n − k) and r = d − (n − 1). Fix C ⊂ [n] k a cluster and G ⊂ C a gauge-fix as in §4.4. Let C \ G = {J 1 , J 2 , . . . , J r } and denote x i := ∆ J i . Let T (C, G) ≃ (C × ) r be the subtorus of T (C) := Spec(C[∆ I | I ∈ C]) satisfying ∆ J = 1 for all J ∈ G. Let Gr(k, n) G ⊂ Gr(k, n) be the subspace where ∆ J = 1 for all J ∈ G. We have a rational map π C,G : T (C, G) → Gr(k, n) G induced by the positive parametrization T (C) → Gr(k, n) in §4.3. We identify Gr(k, n) G birationally with Conf(k, n). The following result follows from (4.1) and Proposition 4.3.
Proposition 11.3. The map π C,G is birational, the restriction to R d−(n−1) >0 is a homeomorphism onto Conf(k, n) >0 , and every Plücker coordinate ∆ I pulls back under π C,G to a positive Laurent polynomial ∆ I (x) in x 1 , . . . , x r .

A Laurent monomial
in the Plücker coordinates is T -invariant if it has weight 0, i.e., I a I e I = 0. For fixed (C, G), each Laurent monomial M (11.1) pulls back to a subtraction-free rational function M(x) by Proposition 11.3. We say that M is nearly convergent with respect to (C, G) if M(x) is nearly converegent.
Lemma 11.4. Let M be a T -invariant monomial and let (C, G) and (C ′ , G ′ ) be two choices of cluster and gauge-fix. Then M is nearly convergent with respect to (C, G) if and only if it is nearly convergent with respect to (C ′ , G ′ ).
Proof. Pulling M back to T (C) we have the notion of M being nearly convergent with respect to C. The T -invariance of M implies that M is nearly convergent with respect to C if and only if it is nearly convergent with respect to some (C, G). The positive parametrizations of two clusters C and C ′ are related by invertible subtraction-free rational transformations, and this implies that the notion of nearly convergent does not depend on cluster.
11.5. Let P (x) = P (x 1 , . . . , x r ) = I ∆ I (x 1 , . . . , x r ) denote the product of all the Plücker variables, considered as a Laurent polynomial in x 1 , . . . , x r . Let P (k, n) = N[P (x)] ⊂ R r be the Newton polytope of P (x). Thus P (k, n) is the Minkowski sum of the Newton polytopes N[∆ I (x)]. By Theorem 10.3 (specifically, the equivalence of the secondary fan structure and the positive fan structure) and Lemma 11.1, we have the following.
Proposition 11.5. There is a bijection F →∆(F ) between the set of faces {F ⊂ P (k, n)} of P (k, n) and the set D(k, n) of regular subdivisions of the hypersimplex into positroid polytopes.

Γ = {M | M is T -invariant and nearly convergent}
denote the (finitely-generated) monoid of nearly convergent, T -invariant, Laurent monomials in ∆ I . Let C[Γ] ⊂ C(Gr(k, n)) T be the subring of T -invariant rational functions on the Grassmannian generated by M ∈ Γ. We will also identify C[Γ] with the subring C[M(x) | M ∈ Γ] ⊂ C(x) of rational functions on the torus T (C, G).
The following result follows from Lemma 11.2 and the fact that each variable x i is the image in C(x) of some T -invariant monomial M. See also [AHLa,Section 10].
Lemma 11.6. The ring C[Γ] ⊂ C(x) is spanned by the rational functions x v /P (x) ℓ for an integer ℓ ≥ 1 and v ∈ ℓ P (k, n) ∩ Z r .
Lemma 11.7. Let f ∈ C[Γ]. Then there exists a constant D ∈ R such that |f (x)| ≤ D for all x ∈ R r >0 . Proof. By Lemma 11.6, it is enough to show that x v /P (x) ℓ is bounded above on R r >0 , where v is a lattice point in the polytope ℓP (k, n). We may write cv = i c i v i where v i are the vertices of ℓ P (k, n) and c, c i are nonnegative integers satisfying c > 0 and c = i c i . Then 11.7. For a lattice polytope P ⊂ R r , one has a proper normal toric variety X P which depends only on the normal fan of P [Ful]. The toric variety X P contains a dense torus T = (C × ) r ⊂ X P . The subspace X P,>0 := R r >0 ⊂ X P (R) is called the positive part of X P and its closure X P,≥0 is called the nonnegative part of X P . The torus orbits of T on X P stratify X P and X P,≥0 , and the strata are in bijection with faces of P . We have a stratification-preserving homeomorphism between X P,≥0 and the polytope P with its face stratification. For a face F ⊂ P , let X F denote the corresponding closed stratum, which is itself isomorphic to the toric variety for the polytope F . Thus the closed (resp. relatively open) faces of X P,≥0 are exactly the X F,≥0 (resp. X F,>0 ) as F varies over faces of P .
(1) The variety Spec(C[Γ]) is isomorphic to an affine open subset X ′ P (k,n) of the projective (and normal) toric variety X P (k,n) associated to the normal fan of P (k, n). (2) The affine open subset X ′ P (k,n) contains the nonnegative part X P (k,n),≥0 .
vanishes identically on X F,≥0 if and only if v / ∈ ℓF . The subspace X F,≥0 is cut out of X ′ P (k,n) by the vanishing of such functions. Proof. (1) is proven in [AHLa, Section 10]; we sketch the main idea. Recall that a fulldimensional lattice polytope Q is called very ample if for sufficiently large integers s > 0, every lattice point in rQ is a sum of s (not necessarily distinct) lattice points in Q. Given Q, it is known that some dilate ℓQ of Q is very ample. We then have a projective embedding of X Q given by the closure of the image of the map We apply this construction to X P (k,n) , supposing that ℓP (k, n) is very ample. Let P (x) ℓ = m i=0 c i x vm , where ℓP (k, n) ∩ Z r = {v 0 , . . . , v m }. Consider the linear section H = {c 0 y 0 + · · · + c m y m = 0} ⊂ P m , where y i are the homogeneous coordinates on P m . The complement X ′ P (k,n) := X P (k,n) \H is an open subset of X P (k,n) that is an affine variety. The coordinate ring C[X ′ P (k,n) ] is generated by the images of y i /(c 0 y 0 +· · ·+c m y m ), that is, x v i /P (x) ℓ . By Lemma 11.6, this gives the isomorphism Spec(C[Γ]) ≃ X ′ P (k,n) , establishing (1).
Since all the coefficients of P (x) are positive, the linear section does not intersect X P (k,n),≥0 , giving (2).
The torus orbit closure X F is cut out from X P by the vanishing of the homogeneous coordinates {x v | v / ∈ F ∩ Z r } in the embedding (11.2). This gives (3).
12. Topology of nonnegative configuration space 12.1. We continue the notation of §11.3. In particular, a cluster C ⊂ [n] k and a gauge-fix G ⊂ C are fixed. Since Conf(k, n) ≃Gr(k, n)/T , each T -invariant monomial extends to a rational function on Ch(k, n). Let Ch(k, n) ⊂ Ch(k, n) denote the locus where all nearly convergent monomials are regular i.e. where the polar locus of each M ∈ Γ has been removed. Then we have a natural morphism (12.1) ϕ : Ch(k, n) −→ Spec(C[Γ]) = X ′ P (k,n) . It follows from Lemma 11.7 that Ch(k, n) ≥0 ⊂ Ch(k, n). Recall the bijection F →∆(F ) in Proposition 11.5.
is nonnegative on R r and strictly positive on the cone C(F ) in the fan Dr(k, n) >0 . Considering f (x) as a T -invariant rational function on Gr(k, n), we conclude that val(f (V (t))) > 0. Thus f (x) vanishes at X and by Proposition 11.8 (3), ϕ(X) ∈ X F,≥0 . The same argument shows that we must actually have ϕ(X) ∈ X F,>0 .
12.2. We can now prove Theorem 6.2.
It remains to argue that the map ϕ is injective. A point X = r i=1 X i ∈ Ch(k, n) ≥0 is determined by the torus orbit closures X i = T · V i where V i ∈ Π M i ,>0 /T and P M 1 , . . . , P Mr are positroid polytopes that form a regular polyhedral subdivision of the hypersimplex. The point X is uniquely determined by the collection of points {V 1 , . . . , V r }. Thus, the injectivity of ϕ follows from the following proposition which completes the proof of Theorem 12.2.
Proof. Note that for f ∈ C[Γ], we have f (V i ) = f (V j ) whenever f makes sense on both V i and V j . Let us first consider the special case where M = [n] k , and suppose V ∈ Gr(k, n) >0 /T >0 . By Lemma 11.6, the function x v /P (x) ℓ is nearly convergent when v ∈ ℓP (k, n). For sufficiently large ℓ, we can find lattice points v and v + e i inside ℓP (k, n), and the ratio of x v P (x) ℓ and x v+e i P (x) ℓ is equal to x i , so the value of x i (V ) is determined by f (V ), for f ∈ Γ. (Note that the ratios x v P (x) ℓ are always positive because all Plücker variables are positive on V .) By Proposition 11.3, V ∈ Gr(k, n) >0 /T >0 is determined by the positive parameters x 1 , . . . , x r ∈ R >0 , and thus we have recovered the point V . Now suppose that V ∈ Π M,>0 /T >0 where M is an arbitrary connected positroid. Applying Lemma 4.6 we find a cluster C ′ ⊂ M and a gauge-fix G ⊂ C ′ . By Proposition 4.1, we can find a cluster C of [n] k that contains C ′ . We use (C, G) as our positive parametrization of Conf(k, n) >0 , and we suppose that x 1 , . . . , x b belong to C ′ \ G while x b+1 , . . . , x r belong to C \ C ′ . Here, b = dim(M) − (n − 1). To determine V , we need to determine x 1 (V ), . . . , x b (V ). The same argument as in the previous paragraph applies, except we need to ensure that the ratios x v P (x) ℓ and x v+e i P (x) ℓ used do not vanish on Π M,>0 /T >0 . As in the proof of Proposition 12.1, the function x v /P (x) ℓ vanishes on Θ∆ exactly when v does not belong to the face ℓF of ℓP (k, n).
We claim that for each i = 1, 2, . . . , b, the face ℓF contains lattice points v and v + e i for some v. In other words, if y = (y 1 , . . . , y r ) ∈ R r is a vector in the normal cone C(F ) to F then y i = 0 for i = 1, 2, . . . , b. Let p • be a positive tropical Plücker vector such that ∆(p • ) =∆. By Lemma 4.6, there is a unique p ′ • equivalent to p • such that p ′ I = 0 for all I ∈ G, and by Lemma 8.1, it must be the case that p ′ J = 0 for all J ∈ M, since P M is one of the full-dimensional pieces in∆. By Corollary 7.4, setting y i = p ′ I i for I i ∈ C gives a vector in C(F ), and any vector in the relative interior of C(F ) is obtained in this way. All these vectors satisfy y 1 = y 2 = · · · = y b = 0 since p ′ J = 0 for all J ∈ M.
12.3. We end this section with the following question.
Question 12.4. Is the morphism ϕ of (12.1) an isomorphism of algebraic varieties?
We note that Spec(C[Γ]) is a normal variety, while the Chow quotient Ch(k, n) has rather complicated geometry (c.f [KT, Laf]).
13. M 0,n and the case k = 2 13.1. It is well-known [Kap] that we have Conf(2, n) =C onf(2, n) = M 0,n and Ch(2, n) = M 0,n , the moduli space of n points on P 1 and its Deligne-Knudsen-Mumford compactification respectively. The space M 0,n (R) consists of n points z 1 , . . . , z n on a circle S 1 . It is a smooth open manifold with (n−1)!/2 connected components, each of which is diffeomorphic to an open ball of dimension n−3. Each connected component is given as the subspace where the n points are in a fixed dihedral ordering. Fixing such an ordering z 1 < z 2 < · · · < z n (up to dihedral symmetries) we obtain the positive component (M 0,n ) >0 ⊂ M 0,n (R). It is well-known that the closure Conf(2, n) ≥0 = (M 0,n ) ≥0 of (M 0,n ) >0 in M 0,n (R) is homeomorphic as a stratified space to the associahedron A n , agreeing with Theorem 12.2. The affine variety M 0,n := Ch (2, n) sits in between M 0,n and M 0,n . It can be obtained from M 0,n by removing all boundary divisors of M 0,n that do not intersect (M 0,n ) ≥0 in codimension one, see [Bro].
13.2. Let us now spell out our combinatorial constructions in this case. A positroid M of rank 2 on [n] is given by a collection of conditions of the following form: (1) For some i ∈ [n], we have i ∈ I for all I ∈ M.
(2) For some j ∈ [n], we have j / ∈ I for all I ∈ M. , it is not difficult to see that there is a canonical isomorphismΠ M /T ≃ M 0,r that sends Π M,>0 /T >0 to (M 0,r ) >0 , where r is the number of cyclic intervals. If r = 3, then M 0,3 is a point, as expected.
13.3. The faces of the associahedron A n are labeled by planar trees T with n cyclically ordered leaves 1, 2, . . . , n, with internal vertices having degree at least 3. Some such trees for n = 5 are drawn below: Proposition 13.1 ( [Kap]). The map gives a bijection between planar trees with n leaves and regular subdivisions of the hypersimplex ∆(2, n) into positroid polytopes.
These hypersimplex decompositions can be obtained by intersecting ∆(k, n) with the hyperplanes H e := {x a + x a+1 + · · · + x b = 1}, one for each each internal e of T , where for an internal edge e, we set [a, b] to be the cyclic interval of leaves on one side of e. The positroid polytope P M(v) has as interior facets the H e where v is incident to e. If T is the star with a single internal vertex v and no edges then∆ T = {P M(v) = ∆(2, n)} is the trivial decomposition.
13.4. Now let p • ∈ Dr(2, n) >0 be a (finite) positive tropical Plücker vector. We assign a planar tree T (p • ) to p • as follows. For 1 ≤ a < b < c < d ≤ n, we consider which of the following three possibilities holds: (1) p ac + p bd = p ab + p cd < p ad + p bc , (2) p ac + p bd = p ad + p bc < p ab + p cd , (3) p ac + p bd = p ab + p cd = p ad + p bc .
The tree T (p • ) has the property that (1) the shortest path from leaf a to leaf d does not intersect the shortest path from leaf b to leaf c, (2) the shortest path from leaf a to leaf b does not intersect the shortest path from leaf c to leaf d, there is an internal vertex v such that any shortest path between two of the vertices a, b, c, d passes through v, respectively.
Proposition 13.2. The map p • → T (p • ) defines a fan structure on Dr(2, n) that agrees with the ones in §10. For k > 2, we do not know an easy estimate for dim(∆), nor does the "factorization" in (13.1) hold. We will study of the geometry of Θ∆ ,>0 in future work [ALS+]. 13.6. We shall use the following positive parametrization of Conf(2, n):
Let us take n = 5. Then the non-monomial Plücker variables are ∆ 24 = 1 + x 1 , ∆ 25 = 1+x 1 +x 1 x 2 , and ∆ 35 = x 1 +x 1 x 2 . The common domains of linearity of Trop(∆ 24 ) = min(0, X 1 ), Trop(∆ 25 ) = min(0, X 1 , X 1 + X 2 ) and Trop(∆ 35 ) = X 1 + min(0, X 2 ) give the following fan: As an example, let us consider the integer vector (1, −1) lying on the southeast pointing ray, and substitute (x 1 , x 2 ) = (t, 1/t) into (13.2) to obtain The tropical Plücker vector p • = val(∆ • (V (t))) is given by p 34 = 1 and p J = 0 for J = 34. Taking a = (1/2, 1/2, −1/2, −1/2, 1/2) in (8.1), we see that 13.7. Let us fix a cluster C ⊂ [n] 2 . In the case k = 2, the polytope P (2, n) has the special feature that it is simple, and thus every cone of the normal fan F is a simplicial cone. Let us denote the (first integer point on the) rays of F by r ij , corresponding to the tree with a single interior edge separating leaves i + 1, . . . , j from j + 1, . . . , i − 1, i, where (i, j) varies over the diagonals of a polygon with vertices 1, 2, . . . , n. Also write p ij • for the tropical Plücker vector that maps to r ij under Theorem 7.3. As explained in [AHLa], a consequence of the simplicial-ness of F is that the ring C[Γ] has some particularly nice generators. For (i, j) a diagonal of the polygon with vertices 1, 2, . . . , n, define the rational function (2, n), which can be interpreted as a cross ratio of the four points z i , z i+1 , z j , z j+1 on P 1 . The functions u ij have the following special property.
Proposition 13.3 ( [AHLa, AHLb]). The ring C[Γ] of §11 is the subring of C(Gr(k, n)) generated by the u ij . For two diagonals (i, j) and (i ′ , j ′ ), we have , so it is easy to see that it takes nonnegative values on any tropical Plücker vector p • , and thus u ij is nearly convergent. It is not hard to see that all the inequalities from (1.1) are positive linear combinations of Trop(u ij ) for various i, j.
Let G ⊂ C be a gauge-fix. Setting ∆ I = 1 for I ∈ G, it is not difficult to see that there is an invertible monomial transformation between the n(n−3)/2 functions u ij and the functions ∆ J with J / ∈ C \ G. The fan structure on Dr(2, n) >0 is given by intersecting the cones Trop(u ij ) = 0 and Trop(u ij ) > 0 as i, j vary.
For n = 5 and the parametrization (13.2), the u ij functions are which are easily seen to belong to Γ using Lemma 11.2.

Tropical bridge reduction
We will frequently use the following easy result without mention. 14.1. Let q • be a positive tropical Plücker vector. Pick a ∈ Z and i = j ∈ [n]. We define the tropical bridge T γ (a) = T i,j (a) acting on R ( This formula is the tropicalization of (3.4). When j = i + 1, we write T i (a) := T i,i+1 (a).
Remark 14.3. Our proof below gives proofs of Theorem 9.2 and of Proposition 14.2 that are independent of our earlier proof of Theorem 9.2. We will only apply Proposition 14.2 to q • ∈ Dr(k, n) ≥0 that we separately know to be representable. And once Theorem 9.2 is established, Proposition 14.2 follows for arbitrary q • ∈ Dr(k, n) ≥0 .
14.2. We shall show that bridge reduction (Proposition 3.3) holds for positive tropical Plücker vectors.
. Then we show that at least one of the following holds: Then p • is in the image of the map ǫ i : Dr(k, n−1) ≥0 ֒→ Dr(k, n) ≥0 given by It is easy to see that f M satisfies at least one of the three stated conditions in Proposition 14.4. In Case (1), if f M (i) = i then I / ∈ M for all I containing i. Thus p I = ∞ whenever i ∈ I. It is clear that p • is in the image of ǫ i , and that the image of ǫ i lies inside Dr(k, n) ≥0 . Case (2) is similar.
First consider the case f (1) = 2. Define q • by which clearly satisfies the positive tropical Plücker relations.
Proof. By induction on M, we may assume that q • is representable and thus p ′ • := T i (a) · q • ∈ Dr(M) >0 by Proposition 14.2. The assumption f (1) = 2 implies that I / ∈ M for any I containing {1, 2}. It suffices to show that p ′ I = p I for 2 ∈ I and I ∈ M. Let I 2 = 2J := J ⊔ {2} where J ⊂ [3, n]. We claim that for K ⊂ [3, n] such that 2K ∈ M. To show this we proceed by (downward) induction on |K ∩ J|, the case |K ∩ J| = k − 1 being tautological. If |K ∩ J| < k − 1 then by the exchange relation there exists K ′ with 2K ′ ∈ M such that |K ′ ∩ J| = |K ∩ J| + 1 and 14.4. For the remainder of the proof we assume that f (1) = j > 2, and we set f (2) = ℓ > j. (It is possible for ℓ to equal 1+n.) Let the Grassmann necklace of M be (I 1 , I 2 , . . . , I n ). Then I 1 = 12I, I 2 = 2Ij, I 3 = Ijℓ. Define M ′ to be the positroid with bounded affine permutation given by f ′ (1) = ℓ and f ′ (2) = j and f ′ (a) = f (a) for a ∈ [3, n]. We define q • by q J = recursion given below if 1 / ∈ J and 2 ∈ J p J otherwise.
It follows immediately that instances of (1.1) for q • immediately hold whenever all subsets involved either contain 1 or do not contain 2. We now give the formula for q K2 , K ⊂ [3, n] recursively. At every step, we will check that instances of (1.1) of the form (14. 2) q L2a + q L1b = min(q L12 + q Lba , q L1a + q L2b ) hold, where 3 ≤ b < a ≤ n and K = La ⊂ [3, n]. We say that (14.2) is associated to the pair {L2a, L1b} of subsets, which uniquely determines (14.2). (a) If K2 / ∈ M, for example if K < Ij in dominance order, then we already have p K2 = ∞. We then set q K2 = ∞ as well. (14.2) will hold since both sides were already equal to ∞ for p • (and thus also for q • ). (b) If K = Ij, then we set q I2j = ∞. The equation (14.2) holds because the LHS is already ∞ for p • . To see this, observe that for the pair {I2j, I1b} with b < j, we have that I1b ≥ 3 Ijℓ = I 3 ; for the pair {I2j = L ′ 2aj, L ′ 1bj} where b < a, either b < j and L ′ 1bj ≥ 3 L ′ ajk = I 3 , or b > j and L ′ 1bj ≥ 1 L ′ 12a = I 1 . In both cases I1b (resp. L ′ 1bj) does not belong to M. (c) If K2 ∈ M and K1 / ∈ M, then we set q K2 = p K2 . We call the subset K2 of parallel type. Setting K = La, we see that (14.2) holds because p L2a + p L1b = min(p L12 + p Lab , p L1a + p L2b ) = p L12 + p Lab and all the terms on both sides are unchanged in q • .
(d) Suppose finally that K2, K1 ∈ M, and K2 = I2j. We call such minors general type. We inductively assume that the value of q K ′ 2 has been defined for all K ′ 2 < K2. Suppose that we have found 2 < b < a ≤ n such that (14.3) K = La and L1b ∈ M.
(The existence of such a pair (a, b) follows from an exchange relation argument similar to the proof of Lemma 14.5, see [Lam,Lemma 7.11].) We then define Since L2b < L2a, we may assume that q L2b has already been defined.
Lemma 14.6. Equation (14.4) well-defines the value of q K2 , regardless of the choice of a and b.
Proof. Suppose we have two pairs (b < a) and (b ′ < a ′ ) such that (14.3) holds. Case 1: If a = a ′ , we assume that b ′ < b, and compute as follows.
Noting that q L1b + q L1b ′ < ∞, we conclude that (14.4) gives the same result using (b < a) and (b ′ < a). Case 2: We have b = b ′ . Similar to Case 1. Case 3: The four numbers a, b, a ′ , b ′ are distinct. Set K = Maa ′ , so by assumption M1ab ′ , M1ba ′ ∈ M. Let us assume that b < a < a ′ . If b < b ′ < a, then M1ba, M1b ′ a ′ ∈ M as well by (1.1), and we can reduce to Case 1. In the other cases, we may conclude from (1.1) that M1bb ′ , M1aa ′ ∈ M.
This shows that (14.4) for the pair (246, 145) gives the same result as for the pair (246,136). Note that we have used (14.4) for K2 = 245 and K2 = 236, which may assume to hold since they are both less than 246 in dominance order.
The other case b ′ < b < a < a ′ is similar.
We have now completely defined the vector q • .
We first show that q • has the correct support.
Proof. We have Supp(p • ) = M. Let J ∈ M \ M ′ be such that p J < ∞. We must show that q J = ∞. Note that J = K2 must be of general type. Apply (14.4). If q J < ∞, then by induction at least one of the two pairs {L12, Lba} and {L1a, L2b} must be contained in M ′ , while J = K2 is not. This contradicts the fact that M ′ satisfies the 3-term positive exchange relation. A similar argument shows that for J ∈ M ′ we have q J < ∞.
By Lemma 14.8, to prove Lemma 14.7 only need to check relations where the LHS of (1.1) is finite i.e. both terms on the LHS are indexed by elements in M ′ . Lemma 14.9. Suppose that J, K ∈ M ′ appear on the LHS of (1.1), and both J and K are not of general type. Then all four subsets on the RHS of (1.1) are also not of general type.
Proof. Let X be any point in Π M,>0 and a > 0 be the unique value such that X ′ = x i (−a) · X ∈ Π M ′ ,>0 as in [Lam]. We have ∆ J (X) = ∆ J (X ′ ) and ∆ K (X) = ∆ K (X ′ ). But we also have ∆ I (X ′ ) ≤ ∆ I (X) for all I. It follows that on the RHS of the three-term Plücker relation for the matrix X, all subsets S that appear must satisfy ∆ S (X) = ∆ S (X ′ ). In particular, all subsets S that appear in the relation are not of general type.
Thus (1.1) holds whenever the LHS involves q T 2 where T 2 is not of general type. It thus suffices to consider instances of (1.1) involving q T 2 where T 2 is of general type. q S2b + q Sac = min(q S2a + q Sbc , q S2c + q Sab ) where 2 < a < b < c ≤ n and S2b is of general type. Thus S1b ∈ M and we add q S1b < ∞ to both sides. The LHS becomes min(q S2b + q S1a + q Sbc , q S2b + q S1c + q Sab ) = min(q S12 + q Sab + q Sbc , q S1b + q S2a + q Sbc , q S2b + q S1c + q Sab ) which is equal to (RHS of (14.5) +q S1b ), using only instances of (1.1) for q • that we know must hold.
(c) Let us verify relations of the form q S2ac + q S12b = min(q S12a + q S2bc , q S12c + q S2ab ) where 2 < a < b < c ≤ n and S2ac is of general type. Thus S1ac ∈ M and we add q S1ac < ∞ to both sides. The calculation is similar to (b).
We have S2bd, S2ac ∈ M. If one of S1ad, S1cd, or S1bc is in M then (7.2) implies that S1ac ∈ M. So if S2ac is of parallel type, we may assume that S1bd ∈ M. Thus q S1ab + q S1bd < ∞ and the argument reduces to that in Case (d).
Otherwise, we have S2ac of general type. If S1cd ∈ M then (7.2) implies that S1bd ∈ M, a contradiction. If S1ad ∈ M then again the argument reduces to that in Case (d). The remaining cases are very similar, and can be simplified by using the assumption that q S1bd , q S1ad , q S1cd are equal to ∞.
We have now completed the proof of Lemma 14.7. Finally, let us check that indeed p • → q • is a tropical bridge reduction. Proof. Let p ′ • = T i (a) · q • . By induction on M, we may suppose that q • is representable, so Proposition 14.2 applies and p ′ • is a positive tropical Plücker vector. We have that Supp(p ′ • ) = M and p ′ J = p J except when J = K2 > I 2 is of general type. But the recursion we used to define q K2 can also be applied to p ′ • (resp. p • ), so we conclude that the rest of the values of p • and p ′ • agree. This completes the proof of Proposition 14.4. 14.5. Proof of Theorem 9.3. Since Dr(M) >0 has the structure of a rational polyhedral complex, the statement over Z implies the other statements. If M = {I} is a singleton, then Dr({I}) >0 (Z) consists of the vectors p(I, z) • , z ∈ Z given by p(I, z) J = z if I = J, ∞ otherwise.

Connected positroids
15.1. We collect some facts concerning connected positroids here.
Lemma 15.1. Let M be a connected positroid with bounded affine permutation f = f M . Then at least one of the following holds: (1) f (i) ∈ {i, i + 1, i + n − 1, i + n} for some i ∈ [n]; (2) there exists i ∈ [n] so that i < f (i) < f (i + 1) < i + n such that f ′ = f s i is a bounded affine permutation of a connected positroid.
Proof. Suppose that f = f M is a counterexample. Since no i ∈ [n] satisfies (1), we must have some i ∈ [n] such that i < f (i) < f (i + 1) < i + n. We have f ′ (j) = f (j) if j = i, i + 1 and f ′ (i) = f (i + 1) and f ′ (i + 1) = f (i). Let M ′ satisfy f M ′ = f ′ . Write π, π ′ : [n] → [n] for the permutations that are reductions of f, f ′ modulo n. Since (2)  But M is connected so after renaming A and B we must have i, π(i + 1) ∈ A = [j + 1, i] and i + 1, π(i) ∈ B = [i + 1, j] where we assume that j has been chosen so that |B| is minimal.
We now assume that out of all i ∈ [n] satisfying i < f (i) < f (i+1) < i+n we have chosen i so that |B[i]| is minimal. Suppose that f (r) < f (r + 1) for some r, r + 1 ∈ [i + 2, j]. Since π(r), π(r + 1) ∈ B, the minimality assumption on |B| implies that π(r + 1), r, r + 1, π(r) are in order within (the totally ordered cyclic interval) B. Set f ′′ = f s r and let π ′′ be the reduction of f ′′ modulo n. vertex. Thus there is J ∈ C such that i − 1 ∈ J but i − 1 / ∈ J ′ for all J ′ ∈ C ′ := C \ {J}. The set C ′ is a cluster for some connected positroid M ′ on [n] \ {i − 1} (not depending on C) of the same rank as M and all clusters C ′ for M ′ occur in this way. By induction, there exists a gauge-fix and cluster G ′ ⊂ C ′ for M ′ i.e. |G ′ | = n − 1 and span Z {e J | J ∈ C ′ } = {(x 1 , . . . , x n ) ∈ Z n | x i−1 = 0 and k divides x i }.
It follows that G := (G ′ ∪ {J}) ⊂ C := (C ′ ∪ {J}) is a gauge-fix. Suppose (2) holds. Let G ′ ⊂ C ′ ⊂ M ′ be a gauge-fix and a cluster for M ′ . It is possible to add an edge to the planar bipartite graph G(C ′ ) to obtain the planar bipartite graph G(C) for some cluster C of M, see [Lam,Section 7.4]. We have C = C ′ ∪ {J} where J is the face label of the new face in G(C). Thus G ′ ⊂ C is a gauge-fix.