Isotropical Linear Spaces and Valuated Delta-Matroids


In this page we present some computations done for the paper "Isotropical Linear Spaces and Valuated Delta-Matroids" by Felipe Rincon.

Preliminaries

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: 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: 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 our paper.

Dressians and Tropical Spinor Spaces

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]. 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 our paper asserts the following: 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.

The following table presents a complete list of all 35 non-isomorphic even Δ-matroids on a ground set of at most 5 elements, together with their corresponding partial Wick relations and their associated Dressians (or tropical pure spinor spaces). 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.

# # of Bases Bases Wick Relations Dressian
0. 1 0 relations dressian
1. 2 0 12 relations dressian
2. 3 0 12 13 relations dressian
3. 4 0 12 13 23 relations dressian
4. 4 0 12 13 14 relations dressian
5. 4 13 23 14 24 relations dressian
6. 5 0 13 23 14 24 relations dressian
7. 6 0 12 13 23 14 24 relations dressian
8. 6 12 13 23 14 24 34 relations dressian
9. 7 0 12 13 23 14 24 34 relations dressian
10. 8 0 12 13 23 14 24 34 1234 relations dressian
11. 5 0 12 13 14 15 relations dressian
12. 6 13 23 14 24 15 25 relations dressian
13. 7 0 13 23 14 24 15 25 relations dressian
14. 8 0 12 13 23 14 24 15 25 relations dressian
15. 8 12 13 14 24 34 15 25 35 relations dressian
16. 9 0 12 13 14 24 34 15 25 35 relations dressian
17. 9 12 13 23 14 24 34 15 25 35 relations dressian
18. 10 0 12 13 23 14 24 34 15 25 35 relations dressian
19. 8 14 24 34 1234 15 25 35 1235 relations dressian
20. 10 0 12 14 24 34 1234 15 25 35 1235 relations dressian
21. 11 0 12 13 14 24 34 1234 15 25 35 1235 relations dressian
22. 12 0 12 13 23 14 24 34 1234 15 25 35 1235 relations dressian
23. 10 12 13 23 14 24 34 15 25 35 45 relations dressian
24. 11 0 12 13 23 14 24 34 15 25 35 45 relations dressian
25. 12 0 12 13 23 14 24 34 1234 35 1235 45 1245 relations dressian
26. 11 0 13 23 24 34 1234 15 35 1235 45 1245 relations dressian
27. 12 0 12 13 23 24 34 1234 15 35 1235 45 1245 relations dressian
28. 13 0 12 13 23 14 24 34 1234 15 35 1235 45 1245 relations dressian
29. 12 13 23 14 24 34 1234 15 25 35 1235 45 1245 relations dressian
30. 13 0 13 23 14 24 34 1234 15 25 35 1235 45 1245 relations dressian
31. 14 0 12 13 23 14 24 34 1234 15 25 35 1235 45 1245 relations dressian
32. 14 0 12 13 23 14 24 34 1234 25 35 1235 45 1245 1345 relations dressian
33. 15 0 12 13 23 14 24 34 1234 15 25 35 1235 45 1245 1345 relations dressian
34. 16 0 12 13 23 14 24 34 1234 15 25 35 1235 45 1245 1345 2345 relations dressian

Even Δ-matroid subdivisions

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 lineality space. After modding out by this lineality 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 here (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 here (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 Our 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 Our 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 http://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.