The positive Dressian equals the positive tropical Grassmannian
By David Speyer and Lauren K. Williams
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 $f$-vector theorem. 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$P = \{P_I\}_{I \in {[n] \choose k}}$, first studied by Andreas Dress, who called them valuated matroids. Thinking of each tropical Plücker vector $P$ as a height function on the vertices of the hypersimplex $\Delta _{k,n}$, one can show that the Dressian parameterizes regular matroid subdivisions $\mathcal{D}_P$ 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 GrassmannianReference 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.Footnote1
1
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 $\operatorname {Dr}_{k,n}$ is much larger than the tropical Grassmannian $\operatorname {Trop}Gr_{k,n}$ – for example, the dimension of the Dressian $\operatorname {Dr}_{3,n}$ grows quadratically is $n$, while the dimension of the tropical Grassmannian $\operatorname {Trop}Gr_{3,n}$ is linear in $n$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 $P$ is a tropical Plücker vector corresponding to a realizable tropical linear space, $\mathcal{D}_P$ has at most $\frac{(n-c-1)!}{(k-c)!(n-k-c)!(c-1)!}$ interior faces of dimension $n-c$, with equality if and only if all facets of $\mathcal{D}_P$ correspond to series-parallel matroids. We refer to this result as the $f$-vector theorem. 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 $\Delta _{k,n}$, which are precisely those induced by positive tropical Plücker vectors. However, it is also natural to consider the set of all positroidal subdivisions of $\Delta _{k,n}$, whether or not they are regular. In light of the various nice realizability results for positroids, one might hope that all positroidal subdivisions of $\Delta _{k,n}$ are regular. However, this is not the case. In Section 7, we construct a nonregular positroidal subdivision of $\Delta _{3,12}$, based off a standard example of a nonregular mixed subdivision of $9 \Delta _2$. 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 $\mathcal{N}=4$ 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 $\operatorname {Trop}^+Gr_{k+1,n}$ controls the regular positroidal subdivisions of the amplituhedron$\mathcal{A}_{n,k,2} \subset Gr_{k,k+2}$, which was introduced by Arkani-Hamed and Trnka Reference AHT14 to study scattering amplitudes in $\mathcal{N}=4$ 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 $f$-vector theorem. Then in Section 7, we construct a nonregular positroidal subdivision of $\Delta _{3,12}$, 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 $[n]$ denote $\{1,\dots ,n\}$, and $\binom{[n]}{k}$ denote the set of all $k$-element subsets of $[n]$. Given $V\in Gr_{k,n}$ represented by a $k\times n$ matrix $A$, for $I\in \binom{[n]}{k}$ we let $p_I(V)$ be the $k\times k$ minor of $A$ using the columns $I$. The $p_I(V)$ do not depend on our choice of matrix $A$ (up to simultaneous rescaling by a nonzero constant), and are called the Plücker coordinates of $V$.
2.1. The positive Grassmannian and its cells
Each positroid cell $S_{M}$ is indeed a topological cell Reference Pos, Theorem 6.5, and moreover, the positroid cells of $Gr_{k,n}^{\ge 0}$ glue together to form a CW complex Reference PSW09.
As shown in Reference Pos, the cells of $Gr_{k,n}^{\geq 0}$ are in bijection with various combinatorial objects, including decorated permutations$\pi$ on $[n]$ with $k$ anti-excedances, $\vcenter{\img[][6pt][7pt][{\scalebox{-1}[1]{\text{L}}}]{Images/imgd1f0ece8d595253204b01ba52077910b.svg}}$-diagrams$D$ of type $(k,n)$, and equivalence classes of reduced plabic graphs$G$ of type $(k,n)$. 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 $\vcenter{\img[][6pt][7pt][{\scalebox{-1}[1]{\text{L}}}]{Images/imgd1f0ece8d595253204b01ba52077910b.svg}}$-diagram, and an equivalence class of plabic graphs; we will correspondingly refer to positroid cells as $S_{\pi }$,$S_D$, etc.
2.2. Matroid and positroid polytopes
In what follows, we set $e_I \coloneq \sum _{i \in I} e_i \in \mathbb{R}^n$, where $\{e_1, \dotsc , e_n\}$ is the standard basis of $\mathbb{R}^n$.
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 $k\times n$ matrix $A$ gives rise to a matroid $M(A)=([n],\mathcal{B})$, where $\mathcal{B} = \{I \in \binom{[n]}{k} \ \vert \ p_I(A) \neq 0\}$.Positroids are the matroids $M(A)$ associated to $k\times n$ matrices $A$ with maximal minors all nonnegative. We call the matroid polytope $\Gamma _M$ 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 $f$ with real coefficients. We also have a positive version of Definition 3.1.
The Grassmannian $Gr_{k,n}$ is a projective variety which can be embedded in projective space $\mathbb{P}^{\binom{[n]}{k}-1}$, and is cut out by the Plücker ideal, that is, the ideal of relations satisfied by the Plücker coordinates of a generic $k \times n$ matrix. These relations include the three-term Plücker relations, defined below.
The tropical Grassmannian$\operatorname {Trop}Gr_{k,n}$, first studied in Reference SS04Reference HKT06Reference KT06, parameterizes tropicalizations of ordinary linear spaces, defined over the field of generalized Puisseux series $\mathbb{K}$ in one variable $t$, with real exponents. More formally, recall that there is a valuation $\operatorname {val}_{\mathbb{K}}: \mathbb{K} \setminus \{0\} \to \mathbb{R}$, given by $\operatorname {val}_{\mathbb{K}}(c(t)) = \alpha _0$ if $c(t) = \sum c_{\alpha _m} t^{\alpha _m}$, where the lowest order term is assumed to have non-zero coefficient $c_{\alpha _0} \neq 0$. Then $P$ lies in the tropical Grassmannian $\operatorname {Trop}Gr_{k,n}$ if and only if there is an element $A = A(t) \in Gr_{k,n}(\mathbb{K})$ whose Plücker coordinates have valuations given by $P=\{P_I\}$ (see Reference Pay09Reference Pay12 for a proof). We will call elements of $\operatorname {Trop}Gr_{k,n}$realizable tropical linear spaces. The tropical Grassmannian is a proper subset of the Dressian,Footnote3 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 $Dr_{k,n}$ is the subset of $\mathbb{R}^{[n]\choose k}$ where the tropical three-term Plücker relations hold.
3
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 $Dr^+_{k,n}$ is the subset of $\mathbb{R}^{[n]\choose k}$ where the positive tropical three-term Plücker relations hold.
In general, the Dressian $\operatorname {Dr}_{k,n}$ is much larger than the tropical Grassmannian $\operatorname {Trop}Gr_{k,n}$ – for example, the dimension of the Dressian $\operatorname {Dr}_{3,n}$ grows quadratically is $n$, while the dimension of the tropical Grassmannian $\operatorname {Trop}Gr_{3,n}$ is linear in $n$Reference HJJS08. However, the situation for their positive parts is different. The main result of this section is the following.
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 $\operatorname {Web}_{k,n}$ 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 $G_0 = \operatorname {Web}_{k,n}$, we obtain the following parameterization of $\operatorname {Trop}^+ Gr_{k,n}$.
In the case of $G_0 = \operatorname {Web}_{k,n}$, we can easily invert the maps $\Phi \coloneq \Phi _{G_0}$ and $\operatorname {Trop}\Phi = \operatorname {Trop}\Phi _{G_0}$. This was done in Reference SW05; we review the construction here. First, given $i$ and $j$ labeling horizontal and vertical wires of $\operatorname {Web}_{k,n}$ (i.e. $1 \leq i \leq k$ and $k+1 \leq j \leq n$), let
If $(i,j)$ does not correspond to a region of $\operatorname {Web}_{k,n}$, set $K(i,j)\coloneq [k]$.
Definition 3.13 gives a way to label each face of $\operatorname {Web}_{k,n}$ by a (tropical) Laurent monomial in (tropical) Plücker coordinates. This is shown in Figure 2.
It is easy to generalize Example 3.17, obtaining the following result.
4. The positive tropical Grassmannian and positroidal subdivisions
Recall that $\Delta _{k,n}$ denotes the $(k,n)$-hypersimplex, defined as the convex hull of the points $e_I$ where $I$ runs over $\binom{[n]}{k}$. Consider a real-valued function $\{I\} \mapsto P_I$ on the vertices of $\Delta _{k,n}$. We define a polyhedral subdivision $\mathcal{D}_P$ of $\Delta _{k,n}$ as follows: consider the points $(e_I, P_I)\in \Delta _{k,n} \times \mathbb{R}$ and take their convex hull. Take the lower faces (those whose outwards normal vector have last component negative) and project them back down to $\Delta _{k,n}$; this gives us the subdivision $\mathcal{D}_P$. We will omit the subscript $P$ when it is clear from context. A subdivision obtained in this manner is called regular.
Given a subpolytope $\Gamma$ of $\Delta _{k,n}$, we say that $\Gamma$ is matroidal if the vertices of $\Gamma$, considered as elements of $\binom{[n]}{k}$, are the bases of a matroid $M$, i.e. $\Gamma = \Gamma _M$.
Given a subpolytope $\Gamma$ of $\Delta _{k,n}$, we say that $\Gamma$ is positroidal if the vertices of $\Gamma$, considered as elements of $\binom{[n]}{k}$, are the bases of a positroid $M$, i.e. $\Gamma = \Gamma _M$. The positroidal version of Theorem 4.2 was recently proved in Reference LPW20, and independently in Reference AHLS20.
It follows from Theorem 4.3 that the regular subdivisions of $\Delta _{k+1,n}$ consisting of positroid polytopes are precisely those of the form $\mathcal{D}_P$, where $P=\{P_I\}$ is a positive tropical Plücker vector.
5. A new proof that positively oriented matroids are realizable
In 1987, da Silva Reference dS87 conjectured that every positively oriented matroid is realizable. Reformulating this statement in the language of Postnikov’s 2006 preprint Reference Pos, her conjecture says that every positively oriented matroid is a positroid. In 2017, da Silva’s conjecture was proved by Ardila, Rincón and the second author Reference ARW17, using the combinatorics of positroid polytopes. In this section we will give a new proof of the conjecture, using our Theorem 3.9, which we think of as a “tropical version” of da Silva’s conjecture.
Recall that an oriented matroid of rank $k$ on $[n]$ can be specified by its chirotope, which is a function from $[n]^k$ to $\{ -, 0, + \}$ obeying certain axioms Reference BLVS+99. If $M$ is a full rank $k \times n$ real matrix, the function taking $(i_1, i_2, \ldots , i_k)$ to the sign of the minor using columns $(i_1, i_2, \dots , i_k)$ is a chirotope, and the realizable oriented matroids are precisely the chirotopes occurring in this way. Thus, if $M$ represents a point of the totally nonnegative Grassmannian, then $M$ gives a chirotope $\chi$ with $\chi (i_1, i_2, \ldots , i_k) \in \{ 0, + \}$ for $1 \leq i_1 < i_2 < \cdots < i_k \leq n$.
We define a positively oriented matroid to be a chirotope $\chi : [n]^k \to \{ -,0, + \}$ such that $\chi (i_1, i_2, \ldots , i_k) \in \{ 0, + \}$ for $1 \leq i_1 < i_2 < \cdots < i_k \leq n$. Since every positroid gives rise to a positively oriented matroid, to prove da Silva’s conjecture, we need to verify that every positively oriented matroid comes from a positroid, or in other words, is realizable.
Interestingly, although the definitions of “positively oriented matroid” and “positroid” don’t involve tropical geometry at all, there does not seem to be a way to remove the tropical geometry from our proof without making it significantly longer.
6. Finest positroidal subdivisions of the hypersimplex
In this section we show that finest positroidal subdivisions of the hypersimplex $\Delta _{k,n}$ achieve equality in the first author’s $f$-vector theorem.
In particular, the number of facets of $\mathcal{D}_P$ – that is, the number of matroid polytopes of dimension $n-1$ in $\mathcal{D}_P$ – is at most ${n-2 \choose k-1}$.
The graphical matroid $M_{K_4}$ is not a positroid, and all minors of positroids are positroids Reference ARW16, so we have the following corollary.
If $M$ is a matroid on the ground set $[n]$, with matroid polytope $\Gamma _M$, and $I$ and $J$ are disjoint subsets of $[n]$, then the the polytope $\Gamma _{M \backslash I / J}$ is $\Gamma _M \cap \{ z_i =0 : i \in I \} \cap \{ z_j = 1 : j \in J \}$. So we can phrase Corollary 6.4 as
It follows from Proposition 2.5 that in a matroidal subdivision, all facets correspond to connected matroids.
Combining Theorem 3.9, Theorem 6.2, and Theorem 6.6, we now have the following.
7. Nonregular positroidal subdivisions
In this paper we have discussed the positive Dressian, which consists of weight functions on the vertices of the hypersimplex $\Delta _{k,n}$ which induce positroidal subdivisions of $\Delta _{k,n}$; recall that subdivisions induced by weight functions are called regular or coherent. It is also natural to consider the set of all positroidal subdivisions of $\Delta _{k,n}$, whether or not they are regular. (See Reference DLRS10 for background on regular subdivisions.) In this section, we will construct a nonregular positroidal subdivision of $\Delta _{3,12}$, and also make a connection to the theory of tropical hyperplane arrangements and tropical oriented matroids Reference AD09Reference Hor16.
Our strategy for producing the counterexample is as follows. We will start with a standard example of a nonregular rhombic tiling of a hexagon (with side lengths equal to $3$), and extend it to a nonregular mixed subdivision of $9 \Delta _2$; this mixed subdivision gives rise to a dual arrangement $\mathcal{H}$ of $9$tropical pseudohyperplanes in $\mathbb{T} \mathbb{P}^2$. Moreover, the mixed subdivision corresponds, via the Cayley trick, to a polyhedral subdivision of $\Delta _2 \times \Delta _8$. We then map this polyhedral subdivision to a matroidal subdivision of $\Delta _{3,12}$, and analyze the $0$-dimensional regions of $\mathcal{H}$ to show that it is a positroidal subdivision of $\Delta _{3,12}$. Note that Reference HJJS08, Example 4.7 used a similar strategy to encode a nonregular matroidal subdivision of $\Delta _{3,9}$. We give a careful exposition here in order to verify that our subdivision is positroidal.
7.1. The product of simplices and the hypersimplex
Let $I$ be any $k$-element subset of $[n]$ and let $J = [n] \setminus I$. Let $\Pi _I \subset \Delta _{k,n}$ be the convex hull of all points of the form $e_I - e_i + e_j$ for $i \in I$ and $j \in J$; clearly this set of points is in bijection with $I \times J$. The polytope $\Pi _I$ is isomorphic to $\Delta _{k-1} \times \Delta _{n-k-1}$, with vertices in bijection with $I \times J$.$\Pi _I$ has dimension $n-2$ and sits inside $\Delta _{k,n}$, which has dimension $n-1$. We review standard constructions for passing between polyhedral subdivisions of $\Pi _I$ and matroidal subdivisions of $\Delta _{k,n}$. We will be interested in polyhedral subdivisions of $\Pi _I$ all of whose vertices are vertices of $\Pi _I$, and we will take the phrase “subdivision of $\Pi _I$” to include this condition.
In many references, $I$ is standardized to be $[k]$. However, we will want to keep track of how these standard constructions relate to the property of a matroid being a positroid and, for this purpose, it will be important how $I$ sits inside the circularly ordered set $[n]$, so we do not impose a standard choice of $I$.
Given a matroidal subdivision $\mathcal{D}$ of $\Delta _{k,n}$, we can intersect $\mathcal{D}$ with $\Pi _I$ and obtain a polyhedral subdivision $\mathcal{G}_I$ of $\Pi _I$. If $\mathcal{D}$ is regular, so is $\mathcal{G}_I$.
7.2. From subdivisions of $\Pi _I$ to subdivisions of $\Delta _{k,n}$
Following Reference HJS14, Theorem 7 and Remark 8, as well as Reference Rin13, we will explain how to map each convex hull of vertices of $\Pi _I$ to a matroid polytope inside $\Delta _{k,n}$; this will be the matroid polytope of a principal transversal matroid.
Let $X \subseteq I \times J$. We define a polytope $\gamma (X) = \mathrm{Hull}_{(i,j) \in X} (e_I - e_i + e_j) \subseteq \Pi _I$. We also define a bipartite graph $G(X)$ with vertex set $I \sqcup J = [n]$ and an edge from $i \in I$ to $j \in J$ if and only if $(i,j) \in X$.
Associated to the graph $G(X)$ is the principal transversal matroid$\operatorname {Trans}(G(X))$ (see Reference Bru87 and Reference Whi86, Chapter 7), defined as follows: $B$ is a basis of $\operatorname {Trans}(G(X))$ if and only if there is a matching of $I \setminus B$ to $J \cap B$ in the bipartite graph $G(X)$. The matroid $\operatorname {Trans}(G(X))$ is realized by a $k \times n$ matrix $A = A_X$, with rows labeled by $I$ and columns labeled by $[n]$ where:
•
the values $A_{i j}$ for $(i, j) \in X$ (where $i\in I$ and $j\in J$) are algebraically independent,
•
$A_{i j} = 0$ if $(i, j) \not \in X$ (where $i\in I$ and $j \in J$),
•
$A_{i i'}=\delta _{i i'}$ (where $i, i'\in I$).
In terms of polyhedral geometry, the matroid polytope of $\operatorname {Trans}(G(X))$ is the intersection of $\Delta _{k,n}$ with $e_I + \operatorname {Span}_{\mathbb{R}_{\geq 0}} \{ e_j - e_i : (i,j) \in X \}$. Summarizing, we have the following.
If $\mathcal{G}$ is a polyhedral subdivision of $\Pi _I$, then we can apply $\operatorname {Trans}$ to each polytope in $\mathcal{G}$.
We will eventually be studying triangulations of $\Pi _I$, so we will want to focus on the case that $\gamma (X)$ is an $(n-2)$-dimensional simplex.
We will want to know when the matroids in Lemma 7.4 are positroidal. One direction of Lemma 7.6 comes from Reference Mar19, Theorem 6.3.
7.3. From tropical pseudohyperplane arrangements to subdivisions of the product of simplices
We now explain how to go between tropical pseudohyperplane arrangements and subdivisions of $\Pi _I$. This section is based on Reference AD09, which initiated the study of tropical oriented matroids and conjectured that they are in bijection with subdivisions of the product of two simplices. Reference AD09 proved their conjecture in the case of $\Delta _{k-1} \times \Delta _2$, which is all we need here; Reference Hor16 proved their conjecture in general. Consult these sources for more detail.
Let $\mathbb{T}\mathbb{P}^{k-1}$ denote tropical projective space$\mathbb{R}^k\!/\mathbb{R}(1,1,\ldots ,1)$, and let $c\!\!=\!\!(c_1, \ldots , c_k)\!$ be an element of $\mathbb{T} \mathbb{P}^{k-1}$. The tropical hyperplane$H_c$ centered at $c$ is the set of points $(x_1, x_2, \ldots , x_k) \in \mathbb{T} \mathbb{P}^{k-1}$ such that $\min _{1 \leq j \leq k} \{x_j - c_j\}$ is not unique. If $x=(x_1, x_2, \ldots , x_k)$ is any point of $\mathbb{T} \mathbb{P}^{k-1}$, we let $S(H,x)$ be the set of indices $j \in [k]$ at which $x_j - c_j$ is minimized. Figure 4 shows a tropical hyperplane in $\mathbb{T} \mathbb{P}^2$, where the horizontal and vertical coordinates are $x_1-x_3$ and $x_2-x_3$, and each region is labelled with the set $S(H,x)$ for $x$ in that region. An arrangement of $m$ labelled tropical hyperplanes is a list of $m$ tropical hyperplanes in $\mathbb{T} \mathbb{P}^{k-1}$.
A tropical pseudohyperplane$H$ is a subset of $\mathbb{T} \mathbb{P}^{k-1}$ which is PL-homeomorphic to a tropical hyperplane. Note that the quantity $S(H,x)$ makes sense for $H$ a tropical pseudohyperplane in $\mathbb{T} \mathbb{P}^{k-1}$ and $x \in \mathbb{T} \mathbb{P}^{k-1}$. An arrangement of $m$ labelled tropical pseudohyperplanes is a list of $m$ tropical pseudohyperplanes which intersect in “reasonable” ways, see Reference Hor16, Section 5 for details. Our main focus in this section will be on the case of tropical pseudohyperplanes in $\mathbb{T} \mathbb{P}^2$.
Consider an arrangement of $n-k$ tropical pseudohyperplanes $H_1$,$H_2$, …, $H_{n-k}$ in $\mathbb{T} \mathbb{P}^{k-1}$. Given a point $x \in \mathbb{T} \mathbb{P}^{k-1}$, we define a subset $X(x)$ of $[k] \times [n-k]$ where $(i,j) \in X(x)$ if and only if $j \in S(H_i, x)$. We can thus associate to each $x\in \mathbb{T} \mathbb{P}^{k-1}$ a polytope $\gamma (X(x)) \subseteq \Delta _{k-1} \times \Delta _{n-k-1}$, as well as the matroid polytope $\Gamma _{\operatorname {Trans}(G(X(x)))}$ of the transversal matroid $\operatorname {Trans}(G(X(x)))$. If we let $x$ range over the bounded regions of the tropical pseudohyperplane arrangement, we obtain the interior regions of a subdivision of $\Delta _{k-1} \times \Delta _{n-k-1}$. Using Reference DS04, Theorem 1 and Reference Hor16, Theorems 1.2 and 1.3, this subdivision is regular if and only if tropical pseudohyperplane arrangement can be realized by genuine tropical hyperplanes.
7.4. Our counterexample
We start with the mixed subdivision of $9 \Delta _2$ shown in Figure 5. The subdivision of the central hexagon (with each side of length $3$) is a standard example of a nonregular subdivision of a hexagon into rhombi, originally found by Richter-Gebert, see Reference ER96, Figure 9. Thus, this mixed subdivision of $9 \Delta _2$ is not regular.
Mixed subdivisions of $b \Delta _{a-1}$ are dual to arrangements of $b$ labeled tropical pseudohyperplanes in $\mathbb{T} \mathbb{P}^{a-1}$. The arrangement of $9$ tropical pseudohyperplanes in $\mathbb{T} \mathbb{P}^2$ which is dual to the mixed subdivision from Figure 5 is shown in Figure 6. In this figure we have labeled the coordinates of $\mathbb{T} \mathbb{P}^2$ by $\{ 4, 8, 12 \}$ – placing the labels at the “ends” of the rays, according to which coordinate is becoming large along the ray – and labelled the tropical pseudohyperplanes by $\{ 1,2,3,5,6,7,9,10,11 \}$, placing the label at the trivalent point.
Also, by the “Cayley trick” Reference HRS00Reference San05, mixed subdivisions of $b \Delta _{a-1}$ correspond to polyhedral subdivisions of $\Delta _{a-1} \times \Delta _{b-1}$, with regular mixed subdivisions of $b \Delta _{a-1}$ corresponding to regular polyhedral subdivisions of $\Delta _{a-1} \times \Delta _{b-1}$. Therefore the mixed subdivision from Figure 5 corresponds to a nonregular polyhedral subdivision of $\Pi _{\{ 4,8,12 \}} \subset \Delta _{3,12}$.
It remains to check that this subdivision is positroidal. We need to check that each of the $45$ two-dimensional polytopes in Figure 5, or equivalently, each of the $45$ zero-dimensional cells of the tropical pseudohyperplane arrangement in Figure 6, corresponds to a positroid. Letting $x$ be one of these zero dimensional cells, we must check that $G(X(x))$ is a tree in each case, which can be embedded in a disk as in Lemma 7.6.
For example, let $x$ be the crossing which is circled in Figure 6; the dual rhombus is shaded in Figure 5. We have
8. Appendix. Combinatorics of cells of the positive Grassmannian
In Reference Pos, Postnikov defined several families of combinatorial objects which are in bijection with cells of the positive Grassmannian, including decorated permutations, and equivalence classes of reduced plabic graphs. Here we review these objects as well as parameterizations of cells.
For example, $\pi = (3,\underline{2},5,1,6,8,\overline{7},4)$ has a loop in position $2$, and a coloop in position $7$. It has three anti-excedances, in positions $4, 7, 8$. We let $k(\pi )$ denote the number of anti-excedances of $\pi$.
Postnikov showed that the positroids for $Gr_{k,n}^{\ge 0}$ are indexed by decorated permutations of $[n]$ with exactly $k$ anti-excedances Reference Pos, Section 16.
All perfect orientations of a fixed plabic graph $G$ have source sets of the same size $k$, where $k-(n-k) = \sum \mathrm{color}(v)\cdot (\deg (v)-2)$. Here the sum is over all internal vertices $v$,$\mathrm{color}(v) = 1$ for a black vertex $v$, and $\mathrm{color}(v) = -1$ for a white vertex; see Reference Pos. In this case we say that $G$ is of type$(k,n)$.
As shown in Reference Pos, Section 11, every perfectly orientable plabic graph gives rise to a positroid as follows. (Moreover, every positroid can be realized in this way.)
Each positroid cell corresponds to a family of reduced plabic graphs which are related to each other by certain moves; see Reference Pos, Section 12. From a reduced plabic graph $G$, we can read off the corresponding decorated permutation $\pi _G$ as follows.
We now explain how to parameterize elements of positroid cells using perfect orientations of reduced plabic graphs.
We will associate a parameter $x_{\mu }$ to each face of $G$, letting $\mathcal{P}_G$ denote the indexing set for the faces. We require that the product $\prod _{\mu \in \mathcal{P}_G} x_{\mu }$ of all parameters equals $1$. A flow$F$ from $I_{\mathcal{O}}$ to a set $J$ of boundary vertices with $|J|=|I_{\mathcal{O}}|$ is a collection of paths and closed cycles in $\mathcal{O}$, all pairwise vertex-disjoint, such that the sources of the paths are $I_{\mathcal{O}} - (I_{\mathcal{O}} \cap J)$ and the destinations of the paths are $J - (I_{\mathcal{O}} \cap J)$.
Note that each directed path and cycle $w$ in $\mathcal{O}$ partitions the faces of $G$ into those which are on the left and those which are on the right of $w$. We define the weight$\operatorname {wt}(w)$ of each such path or cycle to be the product of parameters $x_{\mu }$, where $\mu$ ranges over all face labels to the left of the path. And we define the weight$\operatorname {wt}(F)$ of a flow $F$ to be the product of the weights of all paths and cycles in the flow.
Fix a perfect orientation $\mathcal{O}$ of a reduced plabic graph $G$. Given $J \in {[n] \choose k}$, we define the flow polynomial
This material is based upon work supported by the National Science Foundation under agreement No. DMS-1855135, No. DMS-1854225, No. DMS-1854316 and No. DMS-1854512. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Kelli Talaska, Combinatorial formulas for $\Gamma$-coordinates in a totally nonnegative Grassmannian, J. Combin. Theory Ser. A 118 (2011), no. 1, 58–66, DOI 10.1016/j.jcta.2009.10.006. MR2737184, Show rawAMSref\bib{Talaska2}{article}{
label={Tal11},
author={Talaska, Kelli},
title={Combinatorial formulas for $\Gamma $-coordinates in a totally nonnegative Grassmannian},
journal={J. Combin. Theory Ser. A},
volume={118},
date={2011},
number={1},
pages={58--66},
issn={0097-3165},
review={\MR {2737184}},
doi={10.1016/j.jcta.2009.10.006},
}
Reference [Whi86]
Neil White (ed.), Theory of matroids, Encyclopedia of Mathematics and its Applications, vol. 26, Cambridge University Press, Cambridge, 1986, DOI 10.1017/CBO9780511629563. MR849389, Show rawAMSref\bib{White}{collection}{
label={Whi86},
title={Theory of matroids},
series={Encyclopedia of Mathematics and its Applications},
volume={26},
editor={White, Neil},
publisher={Cambridge University Press, Cambridge},
date={1986},
pages={xviii+316},
isbn={0-521-30937-9},
review={\MR {849389}},
doi={10.1017/CBO9780511629563},
}
The first author was partially supported by NSF grants DMS-1855135 and DMS-1854225. The second author was partially supported by NSF grants DMS-1854316 and DMS-1854512.
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