The slack realization space of a matroid

We introduce a new model for the realization space of a matroid, which is obtained from a variety defined by a saturated determinantal ideal, called the slack ideal, coming from the vertex-hyperplane incidence matrix of the matroid. This is inspired by a similar model for the slack realization space of a polytope. We show how to use these ideas to certify non-realizability of matroids, and describe an explicit relationship to the standard Grassmann-Pl\"ucker realization space model. We also exhibit a way of detecting projectively unique matroids via their slack ideals by introducing a toric ideal that can be associated to any matroid.


Introduction
Realization spaces of matroids are well studied objects [BVS + 93, BS89,Mnë88] which encode not only whether or not the matroid is realizable, but also carry additional information about the structure of the matroid. A realization (or representation) of a rank d + 1 matroid M is a set of vectors in k d+1 which captures its independence structure. Roughly speaking, a realization space is the set of all such choices of vectors. Fundamental questions in the study of realization spaces of matroids include discovering whether or not a given matroid is realizable, determining over which field it is realizable, finding the structure of the set of realizations, and characterizing when realizations exist. A celebrated theorem of Mnëv states that every semialgebraic set defined over the integers is stably equivalent to the realization space of some oriented matroid. That is, realization spaces of matroids can become arbitrarily complicated. In light of this, we aim to connect the combinatorics of the matroid to properties of its realization space.
We generalize a construction of [GMTW] in which they describe a model for the realization space of a polytope using the slack matrix of the polytope. This model gave a new framework for answering questions about the realizability of polytopes. We extend these results to the setting of matroids, creating the beginnings of a dictionary between the combinatorial properties of the matroid and the algebraic description of its realization space.
In Section 2 we introduce the main objects of study, as well as preliminary results and notation. In Section 3 we discuss two models for the realization space of a matroid. One of our main theorems, Theorem 3.8, shows how the two realization space models can be described via a single overarching variety. In Section 4 we show how the slack realization model can be used to determine whether a matroid has a realization over a certain field. We also reframe the tools of final polynomials [BS89] in terms of slack ideals, and show how they can be used to improve computational efficiency of realizability checking. In Section 5 we introduce a toric ideal associated to a matroid and study its relationship to the projective uniqueness of the matroid. In Appendix A we include a table of notation used throughout the paper. The computations in this paper are done in Macaulay2 [GS] with the help of the Matroids package [Che15]; the code we used can be found at http://sites.math.washington.edu/∼awiebe.

The slack matrix of a matroid
Much of this section is analogous to [GMTW,§2] to which we refer the reader for further details and excluded proofs. We assume the reader has familiarity with the basic definitions from matroid theory, see [Oxl11] or [GM12]. Throughout the paper, we assume all matroids are simple (having no loops or parallel elements). Proof. It suffices to show that if we label the rows of S M with [n], the subsets indexing linearly independent rows of S M are the independent sets of M .
if a subset J of E is dependent, then there exists a vector β ∈ k n with support indexed by J such that V β = 0. But now, β S M = (V β) W = 0, so J also indexes a dependent subset of the rows of S M .
Conversely, suppose J indexes a dependent subset of the rows of S M . Then for some β ∈ k n with support indexed by J, we have 0 = β S M = (V β) W . Since W has full rank by Corollary 2.3, it must be the case that V β = 0, so that J also indexes a dependent set of M .
From now on we assume that realizations come with a fixed labelling of ground set elements and hyperplanes, so that two slack matrices of the same matroid cannot differ by permutations of rows and columns. This also allows us to identify hyperplanes of a realization by vectors or the indices of those vectors. Now, we characterize the set of matrices which correspond to slack matrices of a matroid M .
Proof. Suppose S is the slack matrix of some realization of M . Then (i) holds trivially, and (ii) holds by Corollary 2.3. Conversely, suppose S is a matrix satisfying (i) and (ii). By (ii), S has some rank factorization S = AB, where A ∈ k n×(d+1) and B ∈ k (d+1)×h . Let a 1 , . . . , a n ∈ k d+1 be the rows of A and b 1 , . . . , b h ∈ k d+1 be the columns of B. We now recall two equivalence relations on the set of realizations of a matroid M , and illustrate how these equivalences are reflected in slack matrices. For A ∈ GL(k d+1 ), it is easy to check that V and AV define the same matroid. We call these realizations linearly equivalent. If P ∈ k n×n is a permutation matrix which sends i → σ(i), then V and V P define the same matroid up to relabelling the ground set E = [n] with σ(1), . . . , σ(n). Thus if A ∈ GL(k d+1 ) and B is a permutation matrix with any element of k * in the 1's positions, then V, AV B define the same matroid. We call the realizations V, AV B projectively equivalent. Call a matroid M projectively unique (over k) if all realizations are projectively equivalent.  We now define an analog of the slack matrix which can be constructed for any abstract matroid, even ones which are not realizable, as follows.
Definition 2.8. Define the symbolic slack matrix of matroid M to be the matrix S M (x) with rows indexed by elements i ∈ E, columns indexed by hyperplanes H j ∈ H(M ) and (i, j)-entry The slack ideal of M is the saturation of the ideal generated by the (d + 2)-minors of S M (x), namely The symbolic slack matrix of M 4 is in Figure 1. We take the ideal of 4-minors of this matrix, and saturate with respect to the product of all of the variables to get the slack ideal I M 4 . This has codimension 12, degree 293 and is generated by the 72 binomial generators in Table 1. In Section 5 we will see these correspond to the 72 cycles in the bipartite non-incidence graph of this configuration ( Figure 6).
Remark 2.10. In [GMTW], given a set of n vertices V ⊂ k d defining a d-polytope P = conv(V ), they include only the facet defining hyperplanes in the slack matrix.
The triangular prism and its slack matrix as a polytope.
We can also form a matroid associated to this polytope by considering all the hyper- ×n is the matrix obtained from V by appending a 1 to each vector. Then the symbolic slack matrix of P defined in [GMTW] is the restriction of the symbolic slack matrix of matroid M to the subset of columns corresponding to facet-defining hyperplanes. Thus the slack ideal of the polytope is always contained in the slack ideal of the matroid, I P ⊆ I M .
We illustrate with the following example.
Example 2.11. We consider the triangular prism P labelled as in Figure 2. As a 3-polytope, its facets are given by the hyperplanes 1234, 1256, 3456, 135, 246 and the symbolic slack matrix is in Figure 2. Its slack ideal I P is generated by 3 binomials. Considering P as a rank 4 matroid which has the 3 facets 1234, 1256, 3456 of P as its non-bases, we obtain following symbolic slack matrix.
Not only is I P ⊆ I M but in this case I P is the elimination ideal given by eliminating the variables in the columns indexed by the additional hyperplanes H 6 , . . . , H 11 .

Realization space models
A realization space for a rank d + 1 matroid M with n elements is, roughly speaking, a space whose points are in correspondence with (equivalence classes of) collections show that the slack variety defined in the §2 provides such a realization space, and we relate it to another realization space called the Grassmannian of the matroid. Theorem 2.5 characterizes the slack matrices of realizations of a matroid. The next theorem shows that the slack variety captures exactly these matrices.
3, so that its (d + 2)-minors vanish and thus s ∈ V(I M ), as desired.
Since we know that the set of realizations of a matroid is closed under row and column scalings, Theorem 3.1 implies the following corollary. We denote the torus of row and column scalings, Corollary 3.2. The slack variety is closed under the action of the group T n,h , where (k * ) n acts by row scaling (left multiplication by diagonal matrices) and (k * ) h acts by column scaling (right multiplication by diagonal matrices).
Theorem 3.1 and Corollary 3.2 tell us that the slack variety is a realization space for matroid M and the slack variety modulo the action of T n,h is a realization space for the projective equivalence classes of realizations of M .  Proof. Under this embedding, two slack matrices which differ by column scaling are the same point in (P n−1 ) h . So, the result follows by Lemma 2.7.
Next we describe a known model for the realization space of a matroid arising from a subvariety of a Grassmannian. The Grassmannian Gr(d + 1, n) is a variety whose points correspond to (d + 1)-dimensional linear subspaces of a fixed n-dimensional vector space Λ. It embeds into P ( n d+1 )−1 as follows. Any (d + 1)-dimensional linear subspace of Λ can be described as the row space of a (d + 1) × n matrix of rank d + 1. However, two matrices A and B have the same row space when then there is a matrix G ∈ GL(d + 1, Λ) such that A = GB. Thus, to ensure that we have a one-toone correspondence between subspaces and points in the Grassmannian, we record a (d + 1) × n matrix by its vector of (d + 1)-minors. We call this the Plücker vector, and it has coordinates indexed by subsets σ of [n] of size d + 1. The Plücker ideal is the set of all polynomials which vanish on every vector of (d+1)-minors coming from some (d+1)×n matrix. It is generated by the homogeneous quadratic Plücker relations and cuts out the Grassmannian as a variety inside P ( n d+1 )−1 .
If we have a rank d + 1 matroid M = (E, B) with realization V ∈ k (d+1)×n , the Plücker coordinate p σ of V is zero if and only if σ is a dependent set of M . Thus, realizations of M correspond to the subvariety of Gr(d + 1, n) defined by setting the Plücker coordinates of non-bases to 0. This variety is also cut out by an ideal, is the ring with one variable for each basis σ ∈ B. This is the ideal obtained from P d+1,n ⊂ k[p] by setting the variables indexed by non-bases of M to 0.
The points in Gr(M ) correspond to GL(k d+1 ) equivalence classes of (d + 1) × n matrices which realize the matroid M .

Universal realization ideal
Given a matroid M , we now define an ideal whose variety contains pairs (s, q), where q is a Plücker vector and s the nonzero entries of a slack matrix, and both come from the same realization of M . If V is a realization of a rank d+1 matroid M = (E, B), then a slack matrix S M [V ] can be filled with the Plücker coordinates of V , which can be seen from (1). Given a hyperplane H j ∈ H(M ), we record all possible substitutions of Plücker variables for slack variables using a matrix M H j whose rows are indexed by i ∈ E\H j , and whose columns are indexed by subsets J = {j 1 , . . . , j d } ⊂ E with J = H; that is, where sgn(i, J) is the sign of the permutation putting (i, j 1 , . . . , j d ) in increasing order.
Intuitively, insisting that the matrices M H j have rank 1 corresponds to ensuring the columns of the slack matrix are simply scaled versions of the appropriate Plücker coordinates. We now state the main result of this section.
The proof of this theorem requires several preliminary lemmas. We first have the following result on Gröbner bases. (For notation and further details see [CLO15].) Lemma 3.9. Fix an elimination order on k[x, p]. Given two x-homogeneous polynomials f, g and an x- eliminating the x variables. Let G be a Gröbner basis for U M with respect to this ordering. Then, it suffices to show that (2) If we start with a generating set satisfying (2), then by by Lemma 3.9, any terms which are added to G after applying each step of Buchberger's algorithm with xdegree 0 must be the reduction of an S-pair of elements which also have x-degree 0, and are therefore also contained in P M .
It remains to show that an initial generating set of U M satisfies (2). Taking the generating set of the definition, it is enough to show that any minor in I 2 (M H ) not containing a slack variable is already in P M . It is not hard to check that any such minor already arises in P M as some 3-term Plücker relation having a term p σ p τ for some σ / ∈ B which gets set to zero.
Proof of Theorem 3.8.
(i) This follows from the definition of Gr(M ) and Lemma 3.10.
From q we can obtain a (d + 1) × n matrix V with Plücker vector whose nonzero coordinates come from q. We claim that V is a realization of M , so that S  The (i, H) slack entry s iH is of the form det(v i , w 1 , . . . , w d ) for some choice of w 1 , . . . , w d which span the hyperplane H. Each subsequent column of M H has entries det(v i , v j 1 , . . . , v j d ), obtained from q as above, for j 1 < · · · < j d spanning H. Since each v j k lies on hyperplane the H, there is a sequence of elementary column operations that takes the matrix with columns v i , w 1 , . . . , w d to the one with columns v i , v j 1 , . . . , v j d for each i. These column operations change the determinant by some scale factor λ ∈ k * for all i, so that each column of M H is a scalar multiple of the first column of slack entries as required.
Corollary 3.11. Let M = (E, B) be a rank d + 1 matroid. Then By universality [Mnë88], we do not expect that I M is radical for every matroid. Thus, Corollary 3.11 may be the strongest relationship between I M and U M .

Non-realizability
In this section we illustrate how the slack ideal can be used to determine matroid realizability over a given field. This is a well-studied problem [BVS + 93, BS89,Mnë88] for which a complete characterization is only known in a very limited number of cases. Observe that Theorem 3.1 gives us the following criterion for realizability. We now recast this into a test for realizability in terms of the slack ideal.       Figure 4. To simplify the computation, we use Corollary 3.2 and note that we can select a representative of each projective equivalence class by fixing certain variables in the slack matrix to be 1 (see §5.1 for more details). Fixing the variables x 14 , x 27 , x 28 , x 3,12 , x 41 , x 46 , x 47 , x 4,10 , x 57 , x 67 , x 72 , x 73 , x 74 , x 75 , x 77 , x 79 , x 7,11 , x 7,12 , x 84 to 1 and computing the slack ideal I M 8 in Macaulay2 [GS], we find that it is not the unit ideal. However, it contains the polynomial x 2 8,12 + x 8,12 + 1. Since this polynomial has only the complex roots −1±i √ 3 2 , we get by Corollary 4.1 that M 8 is not realizable over R, but it is realizable over C by Proposition 4.2.

Final Polynomials
The method of final polynomials introduced in [BS89, §4.2] certifies when a matroid has no realization. We define an analogous polynomial for the slack ideal, and show how it can be used to improve computational efficiency of checking non-realizability.
Definition 4.5. Let M be any matroid of rank d + 1. Let S be the multiplicatively closed set generated by taking finite products of the variables x in the symbolic slack where I d+2 (S M (x)) is the ideal of (d + 2)-minors of the symbolic slack matrix of M .
We now have the following result, which demonstrates that the existence of slack final polynomials gives a certificate for non-realizability.
Proposition 4.6. Let M be a matroid of rank d + 1. The following are equivalent.
Remark 4.7. Over an algebraically closed field, these conditions are equivalent to the matroid being non-realizable by Proposition 4.2. When k is not algebraically closed, these conditions imply non-realizability, but if a matroid is not realizable there may not be a slack final polynomial.
Example 4.8. Recall that the complex matroid M 8 has a complex realization, as 1 / ∈ I M 8 ⊆ Q[x]. However, by the above proposition, this means that even though M 8 is not realizable over Q, it does not have a slack final polynomial. Proof.
(i) ⇒ (iii) Suppose 1 ∈ I M . Since I M is the saturation of I d+2 (S M (x)), this implies that there exists a monomial m ∈ k[x] such that m · 1 ∈ I d+2 (S M (x)). Then, we observe the m is already a slack final polynomial for M .
(ii) ⇒ (i) If there is a monomial m ∈ k[x] such that m ∈ I d+2 (S M (x)), then after saturation we find 1 ∈ I M .
(iii) ⇒ (ii) Suppose f is a slack final polynomial for M . Since f ∈ (S + I M ), there exists a monomial m and a g ∈ I M with f = m + g. Since g ∈ I M , there exists a monomial n such that ng ∈ I d+2 (S M (x)), so nm = nf − ng ∈ I d+2 (S M (x)) is a monomial in I d+2 (S M (x)).
Remark 4.9. In practice saturation of the ideal I d+2 (S M (x)) can be quite slow, which often makes testing realizability via checking 1 ∈ I M infeasible. Thus the real power of Proposition 4.6 is that one often finds relatively small monomials which are already contained in I d+2 (S M (x)). So, if one simply wants to certify non-realizability, a faster method is to compute I d+2 (S M (x)) and check, for example, if x ∈ I d+2 (S M (x)).
In the following example we exhibit how this method can be useful for certifying non-realizability. Example 4.11. Consider the Vámos matroid pictured in Figure 5. It is a rank 4 matroid M v on 8 elements whose non-bases are given by the sets 1234, 1256, 3456, 3478, and 5678. It is one of the smallest matroids known to be non-realizable over every field. However, the Vámos matroid has 41 hyperplanes, so that its slack matrix is an 8 × 41 matrix containing 200 distinct variables. Even computing the full set of minors of this matrix is computationally impractical. We note though, that it always suffices to show that Proposition 4.6 (ii) holds for some subideal of the ideal of (d + 2)-minors. In particular, we can look at the minors of a submatrix of S Mv (x). Consider the submatrix of the Vámos symbolic slack matrix in Figure 5. One can easily check with Macaulay2 that the monomial given by the product of all the variables in this submatrix is already in the minor ideal of this submatrix (over Q and various finite fields), making M v non-realizable over these fields by Propositions 4.6 and 4.2.

Projective uniqueness of matroids
The simplest slack realization spaces are those belonging to projectively unique matroids. In this case, we know that there is a single realization up to projective transformations; in other words, V(I M ) is the toric variety which is the closure of the orbit of some realization under the action of T n,h . This implies √ I M = I(V(I M )) is a toric ideal; however, universality suggests that I M need not be radical. A natural question which arises is whether projectively unique matroids correspond exactly to matroids with toric slack ideals. To study this question we introduce an intermediate toric ideal associated to a matroid.
Definition 5.1. Define the non-incidence graph of matroid M as the bipartite graph G M with one node for each element of the ground set of M , one node for each hyperplane, and an edge between element i and hyperplane H j if and only if i / ∈ H j . Notice that G M records the support of the slack matrix S M , and so we can think of its edges as being labelled by the corresponding entry of S M (x). (See Figure 6 for an example of the graph G M 4 for the matroid M 4 of Example 2.9, and Figure 7 for the non-incidence graph of the non-Fano matroid.) Let A G be the set of vectors forming the columns of the vertex-edge incidence matrix of the graph G, and let T G be the toric ideal of the vector configuration A G . The toric ideal of a bipartite graph is a well-studied object [OH99,Vil95]. If a matroid M has a slack matrix S M which is a 0-1 matrix, then the toric ideal T G M associated to the graph G M is the ideal of the orbit of S M under the action of the torus T n,h . So, T G M describes one projective equivalence class of slack matrices of M . We now define an analogous toric ideal for any projective equivalence class.
where c+ and c− are alternating edges from the cycle c.
We note that the cycle ideal of a realization provides a way to distinguish projective equivalence classes of realizations of M , as well as detect projective uniqueness. Proof. By Corollary 5.5 and Theorem 3.1, we get V( Then since both varieties are irreducible, the result follows.
In fact, we can have I M = C V for a realization V of M . In this case, call I M cyclic.
Theorem 5.8. If the slack ideal of a matroid is cyclic then M is projectively unique and I M is radical. The converse also holds when k is algebraically closed.
Proof. Suppose that I M is cyclic. Then M is projectively unique by Corollary 5.5 and I M is radical, since it is prime by Theorem 5.3. Conversely, suppose that M is projectively unique and I M is radical. By Proposition 5.7, I(V(I M )) = I(V(C V )), so that I M = C V , as both ideals are radical.
Example 5.9. Recall the matroid M 4 from Example 2.9, whose non-incidence graph G M 4 is displayed in Figure 6. The matroid M 4 has a cyclic slack ideal. Hence I M 4 is a radical slack ideal, and M 4 is projectively unique.
Example 5.10. Recall the Fano plane discussed in Example 4.3. Over F 2 , the 126 binomial generators of I M F found in Example 4.3 correspond to each of the cycles in the graph G M F . Over F 2 this is the projectively unique representation of M F , and this ideal is equal to the cycle ideal of the representation.

Scaled Slack Matrices
From Corollary 3.2 we know that quotienting by the action of T n,h on V(I M ) ∩ (k * ) t gives us a realization space for projective equivalence classes of representations of M .
We now give an explicit way of computing the variety of these equivalence classes. As in [GMTW,Lemma 5.5], we scale rows and columns of a slack matrix via the following lemma, to fix one representative of each projective equivalence class.
Lemma 5.11. Given a realization of a matroid M , we may scale the rows and columns of its slack matrix S M so that it has ones in the entries indexed by the edges in a maximal spanning forest F of the graph G M ; the resulting realization of M is projectively equivalent to the original realization of M .
Definition 5.12. Given a matroid M we can take a symbolic slack matrix and set variables corresponding to edges in a maximal spanning forest F to 1 as in Lemma 5.11 to obtain a scaled symbolic slack matrix. Then, the scaled slack ideal is obtained by taking the (d + 2)-minors of this matrix and saturating with respect to the product of all the variables.
Using the scaled symbolic slack matrix not only allows us to study the projective realization space of M , but also proves to be a useful tool for computations because this matrix will have considerably fewer variables.
Example 5.13. Let M N F be the non-Fano matroid. It is a rank 3 matroid on 7 elements depicted in Figure 7 with its symbolic slack matrix. It differs from the Fano plane by the inclusion of 135 as a basis.
We now show that the non-Fano matroid is projectively unique, and write down a realization from the slack matrix. Let F be the spanning tree of G M N F depicted in Figure 7. We set the corresponding variables x 41 , x 51 , x 22 , x 32 , x 52 , x 13 , x 64 , x 55 , x 06 , x 16 , x 56 , x 66 , x 67 , x 08 , x 69 to 1 in the symbolic slack matrix in Figure 7. Taking the ideal of 4-minors and saturating, we find that the ideal consists of equations of the form x i,j − α i,j , for α i,j ∈ Q, so the configuration is projectively unique over Q. Example 5.14. Consider the Perles configuration M of Figure 8. It is a matroid on 9 elements with hyperplanes given by 0678, 347, 156, 128, 045, 358, 013, 246, 257, 48, 17, 36, 14, 23, 02. Its symbolic slack matrix is shown in Figure 8 and has the matrix S(x) studied in [GMTW, §4.3] as a submatrix.
Since the ideal of S(x) will be contained in the ideal of the whole matrix S M (x), it follows from computation in [GMTW] that M is not realizable over Q. However, it is realizable over R, and computing its scaled slack ideal we find that the slack variety consists of the following matrices, where α is a root of the polynomial α 2 − 3α + 1: