# The positive Dressian equals the positive tropical Grassmannian

## Abstract

The *Dressian* and the *tropical Grassmannian* parameterize abstract and realizable tropical linear spaces; but in general, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces – the *positive Dressian*, and the *positive tropical Grassmannian* (which we introduced roughly fifteen years ago in [J. Algebraic Combin. 22 (2005), pp. 189–210]) – so it is natural to ask how these two positive spaces compare. In this paper we show that the positive Dressian equals the positive tropical Grassmannian. Using the connection between the positive Dressian and regular positroidal subdivisions of the hypersimplex, we use our result to give a new “tropical” proof of da Silva’s 1987 conjecture (first proved in 2017 by Ardila-Rincón-Williams) that all positively oriented matroids are realizable. We also show that the finest regular positroidal subdivisions of the hypersimplex consist of series-parallel matroid polytopes, and achieve equality in Speyer’s * theorem -vector*. Finally we give an example of a positroidal subdivision of the hypersimplex which is not regular, and make a connection to the theory of tropical hyperplane arrangements.

## 1. Introduction

The *tropical Grassmannian*, first studied in Reference HKT06Reference KT06Reference SS04, is the space of *realizable tropical linear spaces*, obtained by applying the valuation map to Puisseux-series valued elements of the usual Grassmannian. Meanwhile the *Dressian* is the space of *tropical Plücker vectors* first studied by Andreas Dress, who called them ,*valuated matroids*. Thinking of each tropical Plücker vector as a *height function* on the vertices of the hypersimplex one can show that the Dressian parameterizes regular matroid subdivisions , of the hypersimplex Reference Kap93Reference Spe08, which in turn are dual to the *abstract tropical linear spaces* of the first author Reference Spe08.

There are positive notions of both of the above spaces. The *positive tropical Grassmannian*, introduced by the authors in Reference SW05, is the space of *realizable positive tropical linear spaces*, obtained by applying the valuation map to Puisseux-series valued elements of the *totally positive Grassmannian* Reference PosReference Lus94. The *positive Dressian* is the space of *positive tropical Plücker vectors*, and it was recently shown to parameterize the regular positroidal subdivisions of the hypersimplex Reference LPW20Reference AHLS20.Footnote^{1}

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Although this result did not appear in the literature until recently, it was anticipated by various people including the first author, Nick Early Reference Ear19a, Felipe Rincón, Jorge Olarte.

In general, the Dressian is much larger than the tropical Grassmannian – for example, the dimension of the Dressian grows quadratically is while the dimension of the tropical Grassmannian , is linear in Reference HJJS08. However, the situation for their positive parts is different. The first main result of this paper is the following, see Theorem 3.9.

We give several interesting applications of Theorem 3.9. The first application is a new proof of the following 1987 conjecture of da Silva, which was proved in 2017 by Ardila, Rincón and the second author Reference ARW17, using the combinatorics of positroid polytopes.

Reformulating this statement in the language of Postnikov’s 2006 preprint Reference Pos, da Silva’s conjecture says that every positively oriented matroid is a *positroid*. We give a new proof of this statement, using Theorem 3.9, which we think of as a “tropical version” of da Silva’s conjecture. Interestingly, although the definitions of positively oriented matroid and positroid don’t involve tropical geometry at all, there does not seem to be an easy way to remove the tropical geometry from our proof without making it significantly longer.

There are two natural fan structures on the Dressian: the *Plücker fan*, and the *secondary fan,* which were shown in Reference OPS19 to coincide. Our second application of Theorem 3.9 is a description of the maximal cones in the positive Dressian, or equivalently, the finest regular positroidal subdivisions of the hypersimplex. The following result appears as Theorem 6.6.

It was shown by the first author in Reference Spe09 that if is a tropical Plücker vector corresponding to a realizable tropical linear space, has at most interior faces of dimension with equality if and only if all facets of , correspond to series-parallel matroids. We refer to this result as the * theorem -vector*. Combining this result with Theorem 6.6 gives the following elegant result (see Corollary 6.7):

Most of our paper concerns the *regular* positroidal subdivisions of which are precisely those induced by positive tropical Plücker vectors. However, it is also natural to consider the set of ,*all* positroidal subdivisions of whether or not they are regular. In light of the various nice realizability results for positroids, one might hope that all positroidal subdivisions of , are regular. However, this is not the case. In Section 7, we construct a nonregular positroidal subdivision of based off a standard example of a nonregular mixed subdivision of , We also make a connection to the theory of tropical hyperplane arrangements and tropical oriented matroids .Reference AD09Reference Hor16.

It is interesting to note that the positive tropical Grassmannian and the positive Dressian have recently appeared in the study of scattering amplitudes in SYM Reference DFGK19Reference AHHLT19Reference HP19Reference Ear19bReference LPW20Reference AHLS20, and in certain scalar theories Reference CEGM19Reference BC19. In particular, the second author together with Lukowski and Parisi Reference LPW20 gave striking evidence that the positive tropical Grassmannian controls the regular positroidal subdivisions of the *amplituhedron* which was introduced by Arkani-Hamed and Trnka ,Reference AHT14 to study scattering amplitudes in SYM.

The structure of this paper is as follows. In Section 2 we review the notion of the positive Grassmannian and its cell decomposition, as well as matroid and positroid polytopes. In Section 3, after introducing the notions of the (positive) tropical Grassmannian and (positive) Dressian, we show that the positive tropical Grassmannian equals the positive Dressian. We review the connection between the positive tropical Grassmannian and positroidal subdivisions in Section 4, then give a new proof in Section 5 that every positively oriented matroid is realizable. We give several characterizations of finest positroidal subdivisions of the hypersimplex in Section 6, and show that such subdivisions achieve equality in the theorem. Then in Section -vector7, we construct a nonregular positroidal subdivision of and make a connection to the theory of tropical hyperplane arrangements and tropical oriented matroids ,Reference AD09Reference Hor16. We end our paper with an appendix (Section 8), which reviews some of Postnikov’s technology Reference Pos for studying positroids.

## 2. The positive Grassmannian and positroid polytopes

Let denote and , denote the set of all subsets of -element Given . represented by a matrix for , we let be the minor of using the columns The . do not depend on our choice of matrix (up to simultaneous rescaling by a nonzero constant), and are called the Plücker coordinates of .

### 2.1. The positive Grassmannian and its cells

Each positroid cell is indeed a topological cell Reference Pos, Theorem 6.5, and moreover, the positroid cells of glue together to form a CW complex Reference PSW09.

As shown in Reference Pos, the cells of are in bijection with various combinatorial objects, including *decorated permutations* on with anti-excedances, -*diagrams* of type and equivalence classes of ,*reduced plabic graphs* of type In Section .8 we review these objects and give bijections between them. This gives a canonical way to label each positroid by a decorated permutation, a and an equivalence class of plabic graphs; we will correspondingly refer to positroid cells as -diagram, , etc. ,

### 2.2. Matroid and positroid polytopes

In what follows, we set where , is the standard basis of .

The dimension of a matroid polytope is determined by the number of connected components of the matroid. Recall that a matroid which cannot be written as the direct sum of two nonempty matroids is called *connected*.

Recall that any full rank matrix gives rise to a matroid where , .*Positroids* are the matroids associated to matrices with maximal minors all nonnegative. We call the matroid polytope associated to a positroid a *positroid polytope*.

## 3. The positive tropical Grassmannian equals the positive Dressian

In this section we review the notions of the tropical Grassmannian, the Dressian, the positive tropical Grassmannian, and the positive Dressian. The main theorem of this section is Theorem 3.9, which says that the positive tropical Grassmannian equals the positive Dressian.

In our examples, we always consider polynomials with real coefficients. We also have a positive version of Definition 3.1.

The Grassmannian is a projective variety which can be embedded in projective space and is cut out by the ,*Plücker ideal*, that is, the ideal of relations satisfied by the Plücker coordinates of a generic matrix. These relations include the three-term Plücker relations, defined below.

The *tropical Grassmannian* first studied in ,Reference SS04Reference HKT06Reference KT06, parameterizes tropicalizations of ordinary linear spaces, defined over the field of generalized Puisseux series in one variable with real exponents. More formally, recall that there is a valuation , given by , if where the lowest order term is assumed to have non-zero coefficient , Then . lies in the tropical Grassmannian if and only if there is an element whose Plücker coordinates have valuations given by (see Reference Pay09Reference Pay12 for a proof). We will call elements of *realizable tropical linear spaces*. The tropical Grassmannian is a proper subset of the *Dressian*,Footnote^{3} which parameterizes what one might call *abstract tropical linear spaces*. Moreover, the Dressian has a natural fan structure, whose cones correspond to the regular matroidal subdivisions of the hypersimplex Reference Kap93, Reference Spe08, Proposition 2.2, see Theorem 4.2. Note that the Dressian is the subset of where the tropical three-term Plücker relations hold.

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Also called the *tropical pre-Grassmannian* in Reference SS04 and named in Reference HJJS08 for Andreas Dress’ work on valuated matroids.

The positive tropical Grassmannian was introduced by the authors fifteen years ago in Reference SW05, and was shown to parameterize tropicalizations of ordinary linear spaces that lie in the totally positive Grassmannian (defined over the field of Puiseux series). The positive tropical Grassmannian lies inside the positive Dressian, which controls the regular positroidal subdivisions of the hypersimplex Reference LPW20, see Theorem 4.3. Note that the positive Dressian is the subset of where the positive tropical three-term Plücker relations hold.

In general, the Dressian is much larger than the tropical Grassmannian – for example, the dimension of the Dressian grows quadratically is while the dimension of the tropical Grassmannian , is linear in Reference HJJS08. However, the situation for their positive parts is different. The main result of this section is the following.

Theorem 3.9 was recently announced in Reference LPW20. It subsequently appeared in independent work of Reference AHLS20.

Before proving Theorem 3.9, we review some results from Reference SW05 which allow one to compute positive tropical varieties.

For the proof of Theorem 3.9 it is convenient to use one particular plabic graph (corresponding to the directed graph from Reference SW05, Section 3), see Figure 1.

Applying Theorem 8.8 to the graph from Figure 1, we have the following result.

In the case of the graph we obtain the following parameterization of , .

In the case of we can easily invert the maps , and This was done in .Reference SW05; we review the construction here. First, given and labeling horizontal and vertical wires of (i.e. and let ),

If does not correspond to a region of set , .