The tropical Grassmannian
Abstract
In tropical algebraic geometry, the solution sets of polynomial equations are piecewiselinear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Gröbner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plücker relations. It parametrizes all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians offer a natural generalization of the space of trees. Their faces correspond to monomialfree initial ideals of the Plücker ideal. The tropical Grassmannian is a simplicial complex glued from 1035 tetrahedra.
1 Introduction
The tropical semiring 10]. These operations satisfy the familiar axioms of arithmetic, e.g. distributivity, with and being the two neutral elements. Tropical monomials represent ordinary linear forms , and tropical polynomials is the set of real numbers augmented by infinity with the tropical addition, which is taking the minimum of two numbers, and the tropical multiplication which is the ordinary addition [
(1) 
represent piecewiselinear convex functions . To compute , we take the minimum of the affinelinear forms for . We define the tropical hypersurface as the set of all points in for which this minimum is attained at least twice, as runs over . Equivalently, is the set of all points at which is not differentiable. Thus a tropical hypersurface is an dimensional polyhedral complex in .
The rationale behind this definition will become clear in Section 2, which gives a selfcontained development of the basic theory of tropical varieties. For further background and pictures see [14, §9]. Every tropical variety is a finite intersection of tropical hypersurfaces (Corollary 2.3). But not every intersection of tropical hypersurfaces is a tropical variety (Proposition 6.3). Tropical varieties are also known as logarithmic limit sets [1], BieriGroves sets [4], or nonarchimedean amoebas [7]. Tropical curves are the key ingredient in Mikhalkin’s formula [9] for planar GromovWitten invariants.
The object of study in this paper is the tropical Grassmannian which is a polyhedral fan in defined by the ideal of quadratic Plücker relations. All of our main results regarding are stated in Section 3. The proofs appear in the subsequent sections. In Section 4 we prove Theorem 3.4 which identifies with the space of phylogenetic trees in [5]. A detailed study of the fan is presented in Section 5. In Section 6 we introduce tropical linear spaces and we prove that they are parametrized by the tropical Grassmannian (Theorem 3.6). In Section we show that the tropical Grassmannian depends on the characteristic of the ground field.
2 The tropical variety of a polynomial ideal
Let be an algebraically closed field with a valuation into the reals, denoted . We assume that 1 lies in the image of and we fix with . The corresponding local ring and its maximal ideal are
The residue field is algebraically closed. Given any ideal
we consider its affine variety in the dimensional algebraic torus over ,
Here . In all our examples, is the algebraic closure of the rational function field and “deg” is the standard valuation at the origin. Then , and if then is the order of vanishing of at .
Every polynomial in maps to a tropical polynomial as follows. If
(2) 
and , then denotes the tropical polynomial in (1).
The following definitions are a variation on Gröbner basis theory [13]. Fix . The weight of a term in (2) is . The initial form of a polynomial is defined as follows. Set . Let be the smallest weight of any term of , so that is a nonzero element in . Define as the image of in . We set . For and this means that the initial form is a polynomial in .
Given any ideal , then its initial ideal is defined to be
Theorem 2.1.
For an ideal the following subsets of coincide:

The closure of the set ;

The intersection of the tropical hypersurfaces where ;

The set of all vectors such that contains no monomial.
The set defined by the three conditions in Theorem 2.1 is denoted and is called the tropical variety of the ideal . Variants of this theorem already appeared in [14, Theorem 9.17] and in [7, Theorem 6.1], without and with proof respectively. Here we present a short proof which is selfcontained.
Proof.
First consider any point in the set (a). For any we have and this implies that the minimum in the definition of is attained at least twice at . This condition is equivalent to not being a monomial. This shows that (a) is contained in (b), and (b) is contained in (c). It remains to prove that (c) is contained in (a). Consider any vector in (c) such that for some . Since the image of the valuation is dense in and the set defined in (a) is closed, it suffices to prove that for some . By making the change of coordinates , we may assume that .
Since contains no monomial and since is algebraically closed, by the Nullstellensatz there exists a point . Let denote the maximal ideal in corresponding to . Let be the set of polynomials in whose reduction modulo is not in . Then is a multiplicative set in disjoint from . Consider the induced map
Let be a minimal prime of the ring on the right hand side. We claim that . Suppose not, and pick with . Then is a zerodivisor in , so we can find such that . Since exists in , this implies which is a contradiction.
Now, implies that is a proper ideal in . There exists a maximal ideal of containing , and, since is algebraically closed, this maximal ideal has the form for some . We claim that and . This will imply and hence complete the proof.
Consider any . By clearing denominators, we get with , and not both and lie in . If , then . Hence contains and hence equals the unit ideal, which is a contradiction. If and then the reduction of modulo does not lie in . This means that and is a unit of , so is the unit ideal. But then is not prime, also a contradiction. This completes the proof. ∎
The key point in the previous proof can be summarized as follows:
Corollary 2.2.
Every zero over of the initial ideal lifts to a zero over of .
By zero of an ideal in we mean a point on its variety in . The notion of (reduced) Gröbner bases is welldefined for ideals and (generic) weight vectors , and, by adapting the methods of [13, §3] to our situation, we can compute a universal Gröbner basis . This is a finite subset of which contains a Gröbner basis for with respect to any weight vector . From part (c) of Theorem 2.1 we derive:
Corollary 2.3.
[Finiteness in Tropical Geometry] The tropical variety is the intersection of the tropical hypersurfaces where .
The following result is due to Bieri and Groves [4]. An alternative proof using Gröbner basis methods appears in [14, Theorem 9.6].
Theorem 2.4.
[BieriGroves Theorem] If is a prime ideal and has Krull dimension , then is a pure polyhedral complex of dimension .
We shall be primarily interested in the case when and . Under this hypothesis, the ideal is said to have constant coefficients if the coefficients of the generators of lie in the ground field . This implies in (1), where . Our problem is now to solve a system of tropical equations all of whose coefficients are identically zero:
(3) 
Here the tropical variety is a subfan of the Gröbner fan of an ideal in .
Corollary 2.5.
If has constant coefficients then is a fan in .
3 Results on the tropical Grassmannian
We fix a polynomial ring in variables with integer coefficients:
The Plücker ideal is the homogeneous prime ideal in consisting of the algebraic relations among the subdeterminants of any matrix with entries in any commutative ring. The ideal is generated by quadrics, and it has a wellknown quadratic Gröbner basis (see e.g. [12, Theorem 3.1.7]). The projective variety of is the Grassmannian which parametrizes all dimensional linear subspaces of an dimensional vector space.
The tropical Grassmannian is the tropical variety of the Plücker ideal , over a field as in Section 2. Theorem 2.1 (c) implies
The ring is known to have Krull dimension . Therefore Theorem 2.4 and Corollary 2.5 imply the following statement.
Corollary 3.1.
The tropical Grassmannian is a polyhedral fan in . Each of its maximal cones has the same dimension, namely, .
We show in Section 7 that the fan depends on the characteristic of if and . All results in Sections 2–6 are valid over any field .
It is convenient to reduce the dimension of the tropical Grassmannian. This can be done in three possible ways. Let denote the linear map from into which sends an vector to the vector whose coordinate is . The map is injective, and its image is the common intersection of all cones in the tropical Grassmannian . Note that vector of length lies in . We conclude:

The image of in is a fan of dimension .

The image of or in is a fan of dimension . No cone in this fan contains a nonzero linear space.

Intersecting with the unit sphere yields a polyhedral complex . Each maximal face of is a polytope of dimension .
We shall distinguish the four objects , , and when stating our theorems below. In subsequent sections less precision is needed, and we sometimes identify , , and if there is no danger of confusion.
Example 3.2.
The smallest nonzero Plücker ideal is the principal ideal . Here is a fan with three fivedimensional cones glued along . The fan consists of three half rays emanating from the origin (the picture of a tropical line). The zerodimensional simplicial complex consists of three points.
Example 3.3.
The tropical Grassmannian is the Petersen graph with vertices and edges. This was shown in [14, Example 9.10].
The following theorem generalizes both of these examples. It concerns the case , that is, the tropical Grassmannian of lines in space.
Theorem 3.4.
The tropical Grassmannian is a simplicial complex known as space of phylogenetic trees. It has vertices, facets, and its homotopy type is a bouquet of . spheres of dimension
A detailed description of and the proof of this theorem will be given in Section 4. Metric properties of the space of phylogenetic trees were studied by Billera, Holmes and Vogtmann in [5] (our corresponds to Billera, Holmes and Vogtmann’s .) The abstract simplicial complex and its homotopy type had been found earlier by Vogtmann [16] and by Robinson and Whitehouse [11]. The description has the following corollary. Recall that a simplicial complex is a flag complex if the minimal nonfaces are pairs of vertices. This property is crucial for the existence of unique geodesics in [5].
Corollary 3.5.
The simplicial complex is a flag complex.
We do not have a complete description of the tropical Grassmannian in the general case and . We did succeed, however, in computing all monomialfree initial ideals for and :
Theorem 3.6.
The tropical Grassmannian is a dimensional simplicial complex with vertices, edges, triangles and tetrahedra.
The proof and complete description of will be presented in Section 5. We shall see that differs in various ways from the tree space . Here is one instance of this, which follows from Theorem 5.4. Another one is Corollary 4.4 versus Proposition 5.5.
Corollary 3.7.
The tropical Grassmannian is not a flag complex.
If is a dimensional linear subspace of the vector space , then (the topological closure of) its image under the degree map is a polyhedral complex in . Such a polyhedral complex arising from a plane in is called a tropical plane in space. Since is invariant under scaling, every cone in contains the line spanned by , so we can identify with its image in . Thus becomes a dimensional polyhedral complex in . For , we get a tree.
The classical Grassmannian is the projective variety in defined by the Plücker ideal . There is a canonical bijection between and the set of planes through the origin in . The analogous bijection for the tropical Grassmannian is the content of the next theorem.
Theorem 3.8.
The bijection between the classical Grassmannian and the set of planes in induces a unique bijection between the tropical Grassmannian and the set of tropical planes in space.
4 The space of phylogenetic trees
In this section we prove Theorem 3.4 which asserts that the tropical Grassmannian of lines coincides with the space of phylogenetic trees [5]. We begin by reviewing the simplicial complex underlying this space.
The vertex set consists of all unordered pairs , where and are disjoint subsets of having cardinality at least two, and
(4) 
We now define as the flag complex with this graph. Equivalently, a subset is a face of if any pair satisfies (4).
The simplicial complex was first introduced by Buneman (see [3, §5.1.4]) and was studied more recently by RobinsonWhitehouse [11] and Vogtmann [16]. These authors obtained the following results. Each face of corresponds to a semilabeled tree with leaves . Here each internal node is unlabeled and has at least three neighbors. Each internal edge of such a tree defines a partition of the set of leaves , and we encode the tree by the set of partitions representing its internal edges. The facets (= maximal faces) of correspond to trivalent trees, that is, semilabeled trees whose internal nodes all have three neighbors. All facets of have the same cardinality , the number of internal edges of any trivalent tree. Hence is pure of dimension . The number of facets (i.e. trivalent semilabeled trees on ) is the Schröder number
(5) 
It is proved in [11] and [16] that has the homotopy type of a bouquet of spheres of dimension . The two smallest cases and are discussed in Examples 3.2 and 3.3. Here is a description of the next case:
Example 4.1.
The twodimensional simplicial complex has vertices, edges and triangles, each coming in two symmetry classes:
Each edge lies in three triangles, corresponding to restructuring subtrees. ∎
We next describe an embedding of as a simplicial fan into the dimensional vector space . For each trivalent tree we first define a cone in as follows. By a realization of a semilabeled tree we mean a onedimensional cell complex in some Euclidean space whose underlying graph is a tree isomorphic to . Such a realization of is a metric space on . The distance between and is the length of the unique path between leaf and leaf in that realization. Then we set
Let denote the image of in the quotient space . Passing to this quotient has the geometric meaning that two trees are identified if their only difference is in the lengths of the edges adjacent to the leaves.
Theorem 4.2.
The closure is a simplicial cone of dimension with relative interior . The collection of all cones , as runs over , is a simplicial fan. It is isometric to the BilleraHolmesVogtmann space of trees.
Proof.
Realizations of semilabeled trees are characterized by the four point condition (e.g. [3, Theorem 2.1], [6]). This condition states that for any quadruple of leaves there exists a unique relabeling such that
(6) 
Given any tree , this gives a system of linear equations and linear inequalities. The solution set of this linear system is precisely the closure of the cone in . This follows from the Additive Linkage Algorithm [6] which reconstructs the combinatorial tree from any point in .
All of our cones share a common linear subspace, namely,
(7) 
This is seen by replacing the inequalities in (6) by equalities. The cone is the direct sum (8) of this linear space with a dimensional simplicial cone. Let . Adopting the convention , for any partition of we define denote the standard basis of
These vectors give the generators of our cone as follows:
(8) 
From the two presentations (6) and (8) it follows that
(9) 
Therefore the cones form a fan in , and this fan has face poset . It follows from (8) that the quotient is a pointed cone.
We get the desired conclusion for the cones by taking quotients modulo the common linear subspace (7). The resulting fan in is simplicial of pure dimension and has face poset . It is isometric to the BilleraHolmesVogtmann space in [5] because their metric is flat on each cone and extended by the gluing relations
We now turn to the tropical Grassmannian and prove our first main result. We shall identify the simplicial complex with the fan in Theorem 4.2.
Proof of Theorem 3.4: The Plücker ideal is generated by the quadrics
The tropicalization of this polynomial is the disjunction of linear systems
Every point on the tropical Grassmannian satisfies this for all quadruples , that is, it satisfies the four point condition (6). The Additive Linkage Algorithm reconstructs the unique semilabeled tree with . This proves that every relatively open cone of lies in the relative interior of a unique cone of the fan in Theorem 4.2.
We need to prove that the fans and are equal. Equivalently, every cone is actually a cone in the Gröbner fan. This will be accomplished by analyzing the corresponding initial ideal. In view of (9), it suffices to consider maximal faces of . Fix a trivalent tree and a weight vector . Then, for every quadruple , the inequality in (6) is strict. This means combinatorially that is a fourleaf subtree of .
Let denote the ideal by the quadratic binomials corresponding to all fourleaf subtrees of . Our discussion shows that . The proof will be complete by showing that the two ideals agree:
(10) 
This identity will be proved by showing that the two ideals have a common initial monomial ideal, generated by squarefree quadratic monomials.
We may assume, without loss of generality, that is a strictly positive vector, corresponding to a planar realization of the tree in which the leaves are arranged in circular order to form a convex gon (Figure 1).
Let be the ideal generated by the monomials for . These are the crossing pairs of edges in the gon. By a classical construction of invariant theory, known as Kempe’s circular straightening law (see [12, Theorem 3.7.3]), there exists a term order on such that
(11) 
Now, by our circular choice of realization of the tree , the crossing monomials appear as terms in the binomial generators of . Moreover, the term order on refines the weight vector . This implies
(12) 
Using we conclude that equality holds in (12) and in (10). ∎
The simplicial complex represented by the squarefree monomial ideal is an iterated cone over the boundary of the polar dual of the associahedron; see [12, page 132]. The facets of are the triangulations of the gon. Their number is the common degree of the ideals , and :
The reduced Gröbner basis of (11) has come to recent prominence as a key example in the FominZelevinsky theory of cluster algebras [8]. Note also:
Corollary 4.3.
There exists a maximal cone in the Gröbner fan of the Plücker ideal which contains, up to symmetry, all cones of .
Proof.
The cone corresponding to the initial ideal (11) has this property. ∎
Corollary 4.4.
Every initial binomial ideal of is a prime ideal.
Proof.
If is a binomial ideal then must satisfy the four point conditions (6) with strict inequalities. Hence . The ideal is radical and equidimensional because its initial ideal is radical and equidimensional (unmixed). for some semilabeled trivalent tree
To show that is prime, we proceed as follows. For each edge of the tree we introduce an indeterminate . Consider the polynomial ring
Let denote the homomorphism which sends to the product of all indeterminates corresponding to edges on the unique path between leaf and leaf . We claim that .
A direct combinatorial argument shows that the convex polytope corresponding to the toric ideal has a canonical triangulation into
Corollary 4.5.
The tropical Grassmannian is characteristicfree.
This means that we can consider the Plücker ideal in the polynomial ring over any ground field when computing its tropical variety. All generators of the initial binomial ideals have coefficients and , so contains no monomial in , even if .
5 The Grassmannian of 3planes in 6space
In this section we study the case and . The Plücker ideal is minimally generated by quadrics in the polynomial ring in variables,
We are interested in the dimensional fan which consists of all vectors such that is monomialfree. The fourdimensional quotient fan sits in and is a fan over the threedimensional polyhedral complex . Our aim is to prove Theorem 3.6, which states that consists of vertices, edges, triangles and tetrahedra.
We begin by listing the vertices. Let denote the set of standard basis vectors in . For each subset of we set
Let denote the set of these vectors. Finally consider any of the tripartitions of and define the vectors
This gives us another set of vectors. All vectors in
Later on, the following identity will turn out to be important in the proof of Theorem 5.4:
(13) 
Lemma 5.1 and other results in this section were found by computation.
Lemma 5.1.
The set of vertices of equals
We next describe all the edges of the tropical Grassmannian .

There are edges like and edges like , for a total of edges connecting pairs of vertices both of which are in . (By the word “like”, we will always mean “in the orbit of, where permutes the indices .)

This class consists of edges like .

Each of the tripartitions gives exactly one edge, like .

There are edges like and edges like , for a total of edges connecting a vertex in to a vertex in .

This class consists of edges like . The intersections of the index triple of the vertex with the three index pairs of the vertex must have cardinalities in this cyclic order.

This class consists of edges like .
Lemma 5.2.
The skeleton of is the graph with the edges above.
Let denote the flag complex specified by the graph in the previous lemma. Thus is the simplicial complex on whose faces are subsets with the property that each element subset of is one of the edges. We will see that is a subcomplex homotopy equivalent to .
Lemma 5.3.
The flag complex has triangles, tetrahedra, fourdimensional simplices, and it has no faces of dimension five or more.
The facets of are grouped into seven symmetry classes:
Facet FFFGG: There are fourdimensional simplices, one for each partition of into three pairs. An example of such a tripartition is . It gives the facet . The tetrahedra contained in these foursimplices are not facets of .
The remaining tetrahedra in are facets and they come in six classes:
Facet EEEE: There are tetrahedra like .
Facet EEFF1: There are tetrahedra like .
Facet EEFF2: There are tetrahedra like .
Facet EFFG: There are tetrahedra like .
Facet EEEG: There are tetrahedra like .
Facet EEFG: There are tetrahedra like .
While is an abstract simplicial complex on the vertices of , it is not embedded as a simplicial complex because relation (13) shows that the five vertices of the four dimensional simplices only span three dimensional space. Specifically, they form a bipyramid with the Fvertices as the base and the Gvertices as the two cone points.
We now modify the flag complex to a new simplicial complex which has pure dimension three and reflects the situation described in the last paragraph. The complex is obtained from by removing the FFFtriangles , along with the tetrahedra FFFG and the fourdimensional facets FFFGG containing the FFFtriangles. In , the bipyramids are each divided into three tetrahedra arranged around the GGedges. The following theorem implies both Theorem 3.6 and Corollary 3.7.
Theorem 5.4.
The tropical Grassmannian equals the simplicial complex . It is not a flag complex because of the missing FFFtriangles. The homology of is concentrated in (top) dimension 3; .
The integral homology groups were computed independently by Michael Joswig and Volkmer Welker. We are grateful for their help.
This theorem is proved by an explicit computation. The correctness of the result can be verified by the following method. One first checks that the seven types of cones described above are indeed Gröbner cones of whose initial ideals are monomialfree. Next one checks that the list is complete. This relies on a result in [7] which guarantees that