This page describes some computations done for the paper Isotropical Linear Spaces and Valuated Delta-Matroids by Felipe Rincón.
An even Δ-matroid is a pair M = (E, B), where E is a finite set and B is a nonempty collection of subsets of E satisfying the following symmetric exchange axiom:
For all A, B ∈ B, and for all a ∈ A Δ B, there exists b ∈ A Δ B such that b ≠ a and both A Δ {a,b} and B Δ {a,b} are in B,
where Δ denotes symmetric difference: X Δ Y := (X \ Y) ∪ (Y \ X). The set E is called the ground set of M, and B is called the collection of bases of M. Even Δ-matroids are a natural generalization of classical matroids; in fact, it is easy to see that matroids are precisely those even Δ-matroids whose bases have all the same cardinality.
If M = (E, B) is an even Δ-matroid and T ⊆ E, the twisting M Δ T is the even Δ-matroid over the ground set E whose collection of bases is B Δ T := {B Δ T : B ∈ B}. We will say that two even Δ-matroids are isomorphic if one can be obtained from the other after a twisting and a relabeling of the elements.
Let R be the set of real numbers, and denote by P(E) the collection of subsets of E. A valuated Δ-matroid on the ground set E is a function p : P(E) → R ∪ {∞} satisfying the following axiom:
For all A, B ∈ P(E), and for all a ∈ A Δ B, there exists b ∈ A Δ B such that b ≠ a and p(A) + p(B) ≥ p(A Δ {a,b}) + p(B Δ {a,b}).
Note that the support supp(p) := {B ∈ P(E) : p(B) ≠ ∞} of a valuated Δ-matroid is the collection of bases of an even Δ-matroid over the ground set E, which justifies the name. It is easy to see that a valuated Δ-matroid is exactly the same as a tropical Wick vector, as described in the paper.
Definition: The Dressian Dr(M) of an even Δ-matroid M is the set of valuated Δ-matroids (tropical Wick vectors) whose support is M.1
The space of all valuated Δ-matroids (tropical Wick vectors) on the ground set [n] := {1, 2, ..., n} is the Δ-Dressian ΔDr(n). It is the disjoint union of the Dressians of all even Δ-matroids on the ground set [n].
Definition: The tropical spinor space TSpin(M) is the set of valuated Δ-matroids (tropical Wick vectors) in Dr(M) that are in the tropical variety defined by the ideal generated by all Wick relations.2
The tropical variety defined by the ideal generated by all Wick relations is the tropical pure spinor space TSpin(n). It is the disjoint union of the tropical spinor spaces of all even Δ-matroids on the ground set [n].
Theorem 4.6 in the paper asserts the following:
Theorem: If n ≤ 5 then the tropical pure spinor space TSpin(n) is equal to the Δ-Dressian ΔDr(n).
Proof. This is equivalent to the statement that for any even Δ-matroid M on a ground set of at most 5 elements, its tropical spinor space TSpin(M) is equal to its Dressian Dr(M). We first computed all possible even Δ-matroids on a ground set of at most 5 elements, getting a list of 35 of them up to isomorphism. Each of these even Δ-matroids M has a corresponding set of "partial Wick relations", obtained by taking all Wick relations and substituting by zero the variables that correspond to non-bases of M. We then used Anders Jensen's software Gfan3 to compute all the spaces TSpin(M) and Dr(M). The space TSpin(M) can be computed as the tropical variety defined by the ideal generated by the partial Wick relations corresponding to M, and the space Dr(M) can be computed as the tropical prevariety defined by these partial Wick relations. In all 35 cases we got that these two spaces were equal, completing the proof of the theorem.
Below is a complete list of all 35 non-isomorphic even Δ-matroids on a ground set of at most 5 elements. Text files containing their corresponding partial Wick relations and associated Dressians (or tropical pure spinor spaces) are available further down. In the column describing the bases, we use 0 to denote the empty set. When writing the Wick relations of an even Δ-matroid M, we use variables of the form "pBe", for B a basis of M. The order of the coordinates in the Dressians agrees with the order of the variables in the Wick relations.
It should be mentioned that Gfan takes tropical addition of two numbers to be their maximum and not their minimum, so in order to be consistent with our min convention, all the rays in these spaces should be changed signs. We refer the reader to the Gfan manual for information on how to interpret these outputs.
There is a natural way to associate a 0/1 polytope to any even Δ-matroid. The resulting polytopes are called even Δ-matroid polytopes; they are precisely those 0/1 polytopes whose edges have the form ± e_i ± e_j, with i ≠ j. We show in our paper that the Dressian Dr(M) of an even Δ-matroid M with associated polytope P is precisely the space of height vectors whose induced regular subdivision on P is a subdivision of P into even Δ-matroid polytopes (called an even Δ-matroid subdivision).
As an example of this, let M be the even Δ-matroid with number 34 in our list. Its Dressian Dr(M) is a pure simplicial 11-dimensional polyhedral fan with a 6-dimensional linearity space. After modding out by this linearity space we get a 5 dimensional polyhedral fan whose f-vector is (1,36,280,960,1540,912). All vectors in this fan induce an even Δ-matroid subdivision of the polytope P associated to M, which is usually known as the 5-demicube. Using the software polymake4, we computed these subdivisions for the vectors in the rays and the vectors in the maximal cones of the fan.
The 36 rays in the fan correspond to the coarsest nontrivial even Δ-matroid subdivisions of P. A complete list of these 36 subdivisions can be found below in the text file subdivisionsrays.txt (the order they are listed matches the order of the rays in Dr(M)). They come in two different isomorphism classes: 16 hyperplane splits of P into 2 polytopes (isomorphic to subdivision #0 in the list), and 20 subdivisions of P into 6 polytopes (isomorphic to subdivision #1 in the list).
The 912 maximal cones in the fan correspond to the finest even Δ-matroid subdivisions of P. A complete list of these 912 subdivisions can be found below in the file subdivisions912.txt (the order they are listed matches the order of the maximal cones in Dr(M)). They come in four different isomorphism classes: 120 of them are isomorphic to subdivision #0 (having 12 facets), 480 of them are isomorphic to subdivision #3 (having 12 facets), 120 of them are isomorphic to subdivision #96 (having 12 facets), and 192 of them are isomorphic to subdivision #98 (having 11 facets).
1 This definition extends the definition of Dressian of a matroid given by Hermann, Jensen, Joswig and Sturmfels in "How to Draw Tropical Planes", Electronic Journal of Combinatorics, 16(2) (2009) R6.
2 This definition extends the definition of Grassmannian of a matroid given by Hermann, Jensen, Joswig and Sturmfels in "How to Draw Tropical Planes", Electronic Journal of Combinatorics, 16(2) (2009) R6.
3 Anders N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties, Available at https://www.math.tu-berlin.de/~jensen/software/gfan/gfan.html.
4 Ewgenij Gawrilow and Michael Joswig, polymake: a framework for analyzing convex polytopes, Polytopes - Combinatorics and Computation (Gil Kalai and Günter M. Ziegler, eds.), Birkhäuser, 2000, pp. 43-74.