Conventions: On this page the tropical sum of two numbers is chosen to be their classical minimum and Gröbner fans are defined using initial forms with terms of minimal weight. Similarly, a subdivision of a polytope is induced from a lifting by taking the lower faces of the lifted polytope.

This page is related to the paper: Herrmann, Jensen, Joswig, Sturmfels: How to Draw Tropical Planes, 2008.

A point on the tropical Grassmannian 3_6 defines a tropical plane in tropical projective space TP5. The coordinates of the point are the Plücker coordinates of the plane. The tropical Grassmannian is a finite union of polyhedral cones. It gets the structure of a polyhedral fan in two different ways:
• as a subcomplex of the secondary fan of the hypersimplex Delta(3,6)
• as a subcomplex of the Gröbner fan of the ideal generated by the classical Grassmann-Plücker relations for Gr(3,6).
The structure induced by the Gröbner fan is a refinement of the structure induced by the secondary fan. In fact with the sedondary fan structure the Grassmannian has f-vector:
(1,65,535,1350,1005)
while with the Gröbner fan structure it has f-vector:
(1,65,550,1395,1035)

Notice: The tropical Grassmannian parameterizes tropicalizations of planes but does not in general parameterize all tropical planes when the dimension of the ambient space gets big enough. They are parameterized by the Dressian Dr(3,n) (defined in How to Draw Tropical Planes). The Dressian comes with a natural polyhedral structure as a subfan of the secondary fan of Delta(3,n).

In the relative interior of a cone in the Grassmannian (with one of the above mentioned fan structures) the combinatorics of the defined tropical plane is fixed. Below the maximal cones in the tropical Grassmannian are listed up to symmetry. It is shown how each cone splits when passing from the polyhedral structure induced by the secondary fan to the one induced by the Gröbner fan. Following the table is an explanation of its columns. The table is valid for all characteristics of the field except 2.

 Secondary cone Gröbner cone Pict. Trees Bvector Orbit size Rvector Orbit size Rays Plücker coordinates P0 T0 0 0 0 0 4 30 4 0 0 30 (2,6,14,16) (0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0) P1 T1 0 0 0 1 3 240 3 0 1 240 (2,14,16,61) (0,0,3,1,2,1,0,1,0,2,1,0,2,0,3,1,3,1,0,0) P2 T2 0 0 1 0 2 360 2 1 1 360 (2,14,31,61) (0,0,3,1,2,1,0,1,0,2,2,0,3,0,4,1,2,2,0,0) P3 T3 0 0 1 0 2 90 2 2 0 90 (2,14,21,31) (1,0,2,0,0,1,0,0,0,0,1,1,1,0,2,0,0,1,0,0) P4 T4 0 0 1 0 2 90 2 2 0 90 (5,14,21,31) (1,0,1,0,0,2,0,0,0,0,1,1,1,0,2,0,0,1,0,0) P5 T5 0 1 0 0 1 180 1 2 1 180 (14,21,31,41) (3,0,2,0,0,2,0,2,1,2,2,3,2,0,4,0,0,3,0,1) P6 T6 1 0 0 0 0 15 0 2 2 45 (21,31,41,50) (4,0,4,0,0,4,0,3,3,3,4,4,4,0,4,0,0,4,0,3)

Column "Picture"

The picture is a schematic drawing of the bounded part of the tropical linear space in TP5. Each vertex in the picture is dual to a face of the hypersimplex subdivision associated to the secondary cone. Such face is the polytope of a matroid on {1,2,3,4,5,6}. The text string attached to each vertex describes its matroid. There are three different kinds of text strings. The notation is as follows:
• "{A,B,C,D}"
where A,B,C and D are subsets of {1,2,3,4,5,6}. The matroid is the graphical matroid given by the multi-graph of a four cycle with edge repeated and named according to the four sets. Notice that the ordering of the sets is unimportant.
Example:
Matroid for "{1,2,346,5}"
 1 2 3 1 2 4 1 2 5 1 2 6 1 3 5 1 4 5 1 5 6 2 3 5 2 4 5 2 5 6
GraphBases
We remind the reader that we go from a graph to a matroid by letting every spanning tree define a basis.
• "[A,B;C,D](E)"
where A, B, C, D and E are subsets of {1,2,3,4,5,6}. The matroid is the graphical matroid given by a four cycle with a diagonal edge. Edges are repeated and named acoording to A, B, C, D and E. Here E corresponds to the diagonal edge while (A,B) and (C,D) correspond to the two paths connecting the two valency three vertices.
Example
Matroid for "[1,26;3,4](5)"
 1 2 3 1 2 4 1 2 6 1 3 6 1 4 6 2 3 4 2 3 5 2 3 6 2 4 5 2 5 6 3 4 6 3 5 6 4 5 6
• "Complete Graph Matroid"
This text string represents a matroid of a complete graph. It only appears once in our drawings where the edges of the complete graph have names as given below.
 1 2 3 1 2 4 1 2 5 1 3 4 1 3 6 1 4 5 1 4 6 1 5 6 2 3 4 2 3 5 2 3 6 2 4 6 2 5 6 3 4 5 3 5 6 4 5 6
Affine point configurationBases

Column "B-vector"

This vector indicates the number hexagons, pentagons, quadrangles, triangles and edges not contained in any 2-face, respectively, in the bounded part of the tropical linear space.

Column "Orbit size"

The tropical Grassmannian Gr(3,6) is invariant under certain permutations of its coordinates. Choosing any permutation of the columns of a 3x6 matrix whose determinants are the Plücker coordinates will keep the tropical Grassmannian fixed. The order of this symmetry group is 720. The list above contains all maximal cones up to this symmetry. The orbit size is the the size of the orbit of the Gröbner/secondary cone under this symmetry.

Column "Plücker coordinates"

A relative interior point of the Gröbner cone. The ordering of the Plücker coordinates are as follows: (p123, p124, p125, p126, p134, p135, p136, p145, p146, p156, p234, p235, p236, p245, p246, p256, p345, p346, p356, p456) .
The point is also a relative interior point of the cone of the secondary fan and thus as a lifting function it induces a subdivsion of the hypersimplex. This subdivision is combinatorially dual to the tropical linear space.

Column "Rays"

The extreme rays of the Gröbner fan have been assigned indices according to the list below. In this list the rays are grouped according to their orbit. In total there are three rays up to symmetry. The "rays" column of the table above contains the indices for the extreme rays of the Gröbner cone of that row. Notice that the fan has a lineality space of dimension 6 and that the coordiantes of the rays should be considered modulo this space.

Column "R-vector"

Each Gröbner cone is spanned by a set of extreme rays in the list below. These rays are grouped according to orbit. The R-vector describes how many rays from each orbit is needed to span the Gröbner cone.

 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 20 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 21 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 22 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 23 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 24 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 25 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 26 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 27 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 28 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 29 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 30 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 31 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 32 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 33 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 35 2 2 1 1 1 0 0 0 0 2 1 0 0 0 0 2 2 2 1 1 36 2 1 2 1 0 1 0 0 2 0 0 1 0 0 2 0 2 1 2 1 37 2 1 1 2 0 0 1 2 0 0 0 0 1 2 0 0 1 2 2 1 38 2 1 0 0 2 1 1 0 0 2 1 0 0 2 2 1 0 0 2 1 39 2 1 0 0 1 0 0 2 2 1 2 1 1 0 0 2 0 0 2 1 40 2 0 1 0 1 2 1 0 2 0 0 1 0 2 1 2 0 2 0 1 41 2 0 1 0 0 1 0 2 1 2 1 2 1 0 2 0 0 2 0 1 42 2 0 0 1 1 1 2 2 0 0 0 0 1 1 2 2 2 0 0 1 43 2 0 0 1 0 0 1 1 2 2 1 1 2 2 0 0 2 0 0 1 44 1 2 2 1 0 0 2 1 0 0 0 0 2 1 0 0 2 1 1 2 45 1 2 1 2 0 2 0 0 1 0 0 2 0 0 1 0 1 2 1 2 46 1 2 0 0 2 0 0 1 1 2 1 2 2 0 0 1 0 0 1 2 47 1 2 0 0 1 2 2 0 0 1 2 0 0 1 1 2 0 0 1 2 48 1 1 2 2 2 0 0 0 0 1 2 0 0 0 0 1 1 1 2 2 49 1 0 2 0 2 1 2 0 1 0 0 2 0 1 2 1 0 1 0 2 50 1 0 2 0 0 2 0 1 2 1 2 1 2 0 1 0 0 1 0 2 51 1 0 0 2 2 2 1 1 0 0 0 0 2 2 1 1 1 0 0 2 52 1 0 0 2 0 0 2 2 1 1 2 2 1 1 0 0 1 0 0 2 53 0 2 1 0 1 0 2 2 1 0 0 2 1 1 0 2 0 2 1 0 54 0 2 1 0 0 2 1 1 0 2 1 0 2 2 1 0 0 2 1 0 55 0 2 0 1 1 2 0 1 2 0 0 1 2 0 1 2 2 0 1 0 56 0 2 0 1 0 1 2 0 1 2 1 2 0 1 2 0 2 0 1 0 57 0 1 2 0 2 0 1 1 2 0 0 1 2 2 0 1 0 1 2 0 58 0 1 2 0 0 1 2 2 0 1 2 0 1 1 2 0 0 1 2 0 59 0 1 0 2 2 1 0 2 1 0 0 2 1 0 2 1 1 0 2 0 60 0 1 0 2 0 2 1 0 2 1 2 1 0 2 1 0 1 0 2 0 61 0 0 2 1 2 1 0 1 0 2 1 0 2 0 2 1 2 1 0 0 62 0 0 2 1 1 0 2 0 2 1 2 1 0 1 0 2 2 1 0 0 63 0 0 1 2 2 0 1 0 1 2 1 2 0 2 0 1 1 2 0 0 64 0 0 1 2 1 2 0 2 0 1 2 0 1 0 1 2 1 2 0 0

Trees

The Trees column contains a link to a row of the table below. Intersecting the tropical plane with each of the 6 classical hyperplanes at infinity we get 6 trees. The leaves of such a tree go off in the remaining 5 infinity directions and get names accordingly. The trees are incoded as text strings. There is one type of tree.
• A caterpillar tree is represented by a string of the form C(ab,...,yz) where the leaves ab and the leaves yz form cherries. The dots describes the leaves in between in order from ab to yz. Here the letters are all names of vertices of the tree, that is, numbers between 1 and 6.
 Secondary cone Tree 1 Tree 2 Tree 3 Tree 4 Tree 5 Tree 6 0 C(25,4,36) C(15,3,46) C(16,2,45) C(26,1,35) C(12,6,34) C(13,5,24) 30 1 C(25,6,34) C(15,3,46) C(26,1,45) C(26,1,35) C(12,6,34) C(15,3,24) 240 2 C(25,6,34) C(15,3,46) C(15,4,26) C(15,3,26) C(12,6,34) C(15,3,24) 360 3 C(25,3,46) C(15,3,46) C(15,2,46) C(15,3,26) C(12,3,46) C(15,3,24) 90 4 C(35,2,46) C(15,3,46) C(15,2,46) C(15,3,26) C(13,2,46) C(15,3,24) 90 5 C(23,5,46) C(15,3,46) C(15,2,46) C(15,3,26) C(23,1,46) C(15,3,24) 180 6 C(23,5,46) C(15,3,46) C(15,2,46) C(15,6,23) C(23,1,46) C(15,4,23) 15

Less generic planes in TP5

The lower dimensional cones of the Gröbner fan follow (up to symmetry).

Dimension 9

 Index Orbit size Rays Plücker coordinates 0 45 (21,41,50) (4,0,4,0,0,4,0,3,3,3,3,4,3,0,3,0,0,3,0,3) 1 180 (14,31,61) (0,0,2,1,2,1,0,1,0,2,2,0,3,0,4,1,2,2,0,0) 2 90 (21,31,41) (3,0,2,0,0,2,0,2,1,2,2,3,2,0,3,0,0,3,0,1) 3 180 (2,31,61) (0,0,3,1,2,1,0,1,0,2,2,0,3,0,3,1,2,2,0,0) 4 360 (2,14,61) (0,0,3,1,2,1,0,1,0,2,1,0,2,0,3,1,2,1,0,0) 5 180 (14,21,31) (1,0,1,0,0,1,0,0,0,0,1,1,1,0,2,0,0,1,0,0) 6 180 (2,14,31) (0,0,1,0,0,0,0,0,0,0,1,0,1,0,2,0,0,1,0,0) 7 60 (5,14,21) (1,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0) 8 120 (2,14,16) (0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0)

Dimension 8

 Index Orbit size Rays Plücker coordinates 0 15 (41,50) (3,0,3,0,0,3,0,3,3,3,3,3,3,0,3,0,0,3,0,3) 1 90 (21,41) (3,0,2,0,0,2,0,2,1,2,1,3,1,0,2,0,0,2,0,1) 2 180 (2,61) (0,0,3,1,2,1,0,1,0,2,1,0,2,0,2,1,2,1,0,0) 3 60 (14,31) (0,0,0,0,0,0,0,0,0,0,1,0,1,0,2,0,0,1,0,0) 4 45 (21,31) (1,0,1,0,0,1,0,0,0,0,1,1,1,0,1,0,0,1,0,0) 5 60 (2,31) (0,0,1,0,0,0,0,0,0,0,1,0,1,0,1,0,0,1,0,0) 6 90 (2,14) (0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0) 7 10 (5,14) (0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

Dimension 7

 Index Orbit size Rays Plücker coordinates 0 30 (41) (2,0,1,0,0,1,0,2,1,2,1,2,1,0,2,0,0,2,0,1) 1 15 (21) (1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0) 2 20 (2) (0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

Dimension 6

 Index Orbit size Rays Plücker coordinates 0 1 () (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)