Unlabeled equivalence for matroids representable over finite fields

Transcription

1 Unlabeled equivalence for matroids representable over finite fields November 16, 2012 S. R. Kingan Department of Mathematics Brooklyn College, City University of New York 2900 Bedford Avenue Brooklyn, NY Abstract Two r n matrices A and A representing the same matroid M over GF (q), where q is a prime power, are projective equivalent representations of M if one can be obtained from the other by elementary row operations and column scaling. Bounds for projective inequivalence are difficult to obtain and are known for only a few special classes of matroids. In this paper we define two matrices A and A to be geometric equivalent if, in addition to row operations and column scaling, column permutations are also allowed. We show that the number of geometric inequivalent representations is at most the number of projective inequivalent representions and we give a polynomial time algorithm for determining if two projective inequivalent representations are geometrically equivalent. Thus, from a computational perspective there is no additional cost to altering the definition of equivalence in this manner. The benefit is that it could lead to a new set of theorems for inequivalence with better bounds. 1 Introduction A rank-r simple matroid M with ground set E is representable over a field F if there is a rank preserving map ϕ : E V (r, F ). If M is representable over F, then we can find a matrix A with entries from F that represents M. We can write the matrix in standard form as A = I r D, where the columns of A correspond to non-zero representative vectors from one-dimensional subspaces of V (r, F ). The results in this paper are limited to matroids representable over a finite field. Following the notation in Oxley 7, we denote a finite field as GF (q), where q is a prime power. The columns of A may be viewed as a subset of the columns of the matrix that represents the projective geometry P G(r 1, q). Informally, a rankr simple GF (q)-representable matroid sits inside the corresponding projective geometry P G(r 1, q) like a graph sits inside the complete graph. Figure 1 shows matrix and geometric representations for P G(2, 2) and P G(2, 3). There is no good way of visualizing higher rank projective geometries. Figure 1: Rank 3 projective geometries The study of the structure of representable matroids is similar to graph structure theory in the sense that many of the theorems that hold for graphs have analogs for representable matroids. For example, the Kuratowski-Wagner excluded-minor characterization for planar graphs is a well-known result established in It states that a graph is planar if and only if it has no minor isomorphic to M(K 5 ) and M(K 3,3 ). Given a class M of matroids that is closed under minors and duality, a matroid that is not in M, but whose every single-element deletion and single-element contraction is in M is called a minimal excluded minor. Tutte proved in 1958 that a matroid is binary if and only if it has no minor isomorphic to U 2,4 (the line with four points) 7, Excluded minor characterizations are also available for matroids representable over GF (3) and GF (4) 7, 6.5.7, Rota has conjectured that the class of 98 Copyright SIAM.

2 GF (q)-representable matroids has a finite list of minimal excluded minors 7, When q 4, representability over a finite field is complicated by the presence of inequivalent representations. Two GF (q)- representable matroids, MA and MB may be isomorphic as matroids, but there may be no way of obtaining matrix A from matrix B by applying elementary row operations, column scaling, or a field automorphism. In such a case, we say A and B are inequivalent representations of the same matroid. This amounts to saying A and B are embedded differently in the projective geometry. For example, the three matrices A, B, C listed below, represent the same matroid W 3 (the 3-whirl) over GF (5). The 3-whirl is obtained from P G(2, 2) by deleting the element in the center. The three matrices are identical as matroids, that is MA = MB = MC, but they are inequivalent representations of W 3. A = B = C = I I I Add the column 1, 1, 1 T to each and we see that A with 1, 1, 1 T represents the matroid F7 (non- Fano), whereas B and C with 1, 1, 1 T, respectively, represent the matroid X 7 in Figure 2. Moreover, there is no column that can be added to A to get a matrix that represents X 7. So, if one happens to consider A as a representation of W 3 over GF (5), then there is no indication that X 7 is a single-element extension of W 3, and the fact that X 7 is representable over GF (5) could conceivably be missed. If the goal is to compute single-element extensions of W 3 over GF (5), then based on the prevailing notion of projective equivalence, one has to keep track of all three representations A, B, and C for W 3. We will see in this paper that three distinct representations are unnecessary. We can get away with fewer distinct representations, if we replace the notion of projective equivalence with an unlabeled version that we call geometric equivalence. Formally, two r n matrices A and A representing the same matroid M over a field F are said to be projective equivalent representations of M, if one can be obtained from the other by elementary row operations or column scaling. Otherwise A and A are not projective equivalent or more concisely projective inequivalent. The matrices A and A are said to be (algebraically) equivalent representations of M, if one can be obtained from the other by elementary row operations, column scaling, or field automorphisms. Otherwise they are called inequivalent representations. For prime fields, projective equivalence and equivalence coincide since prime fields do not have non-trivial automorphisms. For finite fields of order q, where q is a prime power, projective equivalence is a refinement of equivalence. Consequently, a result that holds for projective equivalence automatically holds for equivalence. In matroid structure theory, the notion of projective equivalence is what matters. Let B be a basis of a matroid M with ground set E. Then for every e E B, there is a unique circuit C such that e C B e. This circuit is called the fundamental circuit of e with respect to B. The first paper to discuss equivalence in detail was by Brylawski and Lucas 1. They gave a method for coordinatizing a matroid over a field using fundamental circuits. Subsequently, Whittle refined the concept of equivalence by flagging projective equivalence as important for structural results 9. The reader should note that projective equivalence was called strong equivalence in 9. In 2 strong equivalence was renamed as projective equivalence and the second edition of Oxley s book published in 2011 uses the term projective equivalence 7, p In 4 the author developed a computational approach to equivalence and implemented an algorithm for coordinatizing a matroid over a fixed finite field taking into account projective equivalence. That algorithmic approach and implementation led to the concept of geometric equivalence, which is presented in this paper. 2 Geometric Equivalence We define two r n matrices A and A representing the same matroid M over F as geometric equivalent representations of M, if one can be obtained from the other by elementary row operations, column scaling, or column permutations. Otherwise A and A are geometric inequivalent representations of M. Observe that this definition is similar to the definition of projective equivalence, except that we added column permutations. Geometric equivalence is an unlabeled version of projective equivalence; hence the title of the paper. The three representations of W 3 over GF (5), mentioned earlier, are projective inequivalent representations. But there is a non-singular linear trans- 99 Copyright SIAM.

3 formation of the vector space V (3, 5) followed by an unlabeled column permutation that maps B to C. Specifically, we can convert B to C by the following sequence of operations: swap rows 1 and 3; multiply column 6 by 3; and swap columns 1 and 3 and columns 4 and 5. However, there is no such operation that maps A to B. Thus we say B and C are geometric equivalent representations of W 3 over GF (5), whereas A and B are geometric inequivalent representations of W 3 over GF (5). Besides doing an exhaustive search, which we did, the fact that there is no linear transformation followed by a column permutation that maps A to B can be confirmed by checking single-element extensions. Matrix A has six non-isomorphic singleelement extensions. Let us call them A 7, B 7, C 7, D 7, and the well-known F7 and P 7. Matrix B and C each have seven non-isomorphic single-element extensions, A 7, B 7, C 7, D 7, X 7, Y 7 and P 7 (Figure 2). Figure 2: Single-element extensions of W 3 over GF (5) Thus, in a matroid generation algorithm (over a fixed finite field, in this case GF (5)) one needs to keep track of only two geometric inequivalent representations for W 3 over GF (5) as compared to three projective inequivalent representations. This may seem like a small savings, but bounding the number of projective inequivalent representations is difficult. In the absence of an overall bound, bounds have been determined for subclasses of representable matroids. For example, matroids representable over GF (2) and GF (3) are uniquely representable 7, The class of 3-connected GF (4)-representable matroids have at most two projectively inequivalent representations 6 and 7, , and due to the field automorphism they are uniquely representable with respect to equivalence. The class of 3-connected GF (5)- representable matroids have at most six inequivalent representations 8 and 7, Moreover, for a fixed q 7 there are two examples of infinite families of matroids for which the number of projective inequivalent representations depends on the rank 7, Another approach to bounding the number of projective inequivalent representations is to require the class of matroids to have a certain matroid as a sub-structure. In 5 it is shown that if M is a 3- connected GF (q)-representable matroid with a long line L (line with at least q points) as a restriction, then the number of projective inequivalent representations is bounded by the number of projective inequivalent representations of M restricted to L. Further, suppose M has a large plane P (plane with at least 2q elements) as a restriction, but no long line as a restriction, then the number of projective inequivalent representations is bounded by the number of projective inequivalent representations of M restricted to P. In 2 it is shown that a 3-connected GF (q)-representable matroid with a P G(2, q)-minor is uniquely representable. The next theorem is the main result of this section. It establishes a method for computing the number of geometric inequivalent representations and shows that it is at most the number of projective inequivalent representations. So, any bound that holds for projective equivalence automatically holds for geometric equivalence. Moreover, a polynomial time algorithm is given for computing the geometric inequivalent representations from the set of projective inequivalent representations. As a consequence, there is no significant additional computational cost associated with computing geometric inequivalent representations. The proof of the theorem is based on 7, and Copyright SIAM.

4 Theorem 2.1. Let M be an n-element rank-r GF (q)-representable matroid. The number of geometric inequivalent representations of M over GF (q) is at most the number of projective inequivalent representations. Moreover, the geometric inequivalent representations can be obtained from the projective inequivalent representations using a polynomial time algorithm in n, where the coefficient of n 2 is q r2 r. Proof. Let M be a connected n-element rank-r GF (q)-representable matroid with elements of a basis B labeled as {1,..., r} and the remaining elements labeled {r + 1,..., n}. Construct the matrix I r D where column k of D has ones corresponding to the fundamental circuit of k with respect to B and zeros elsewhere. Observe that D may be viewed as an incidence matrix of a bipartite graph G with the rows labeled by B = {1,..., r} and columns labeled by E B = {r + 1,..., n}. The vertices in the two classes of G are labeled with the elements of B and E B, respectively. The edges of G correspond to the ones in D. Select any spanning tree B G of G and assign the corresponding entries in D arbitrary values from GF (q). Since B G is a basis, we can do row and column scaling to reduce all these arbitrary values to one. So, without loss of generality, we may assume these entries are one and the remaining non-zero entries of D are members GF (q), say (a, b, c,...). Call the resulting matrix D. Note that the order in which the remaining entries are labeled by the letters is not important, however, once labeled, they are fixed and give rise to an ordered sequence. The values of (a, b, c,...) may be found by setting up a system of equations using the circuits of M and solving the system over GF (q). If the system has a solution, then the matroid is representable over GF (q). Otherwise it is not representable over GF (q). If there is just one sequence (a, b, c,...) over GF (q), then M is uniquely representable over GF (q) with respect to projective equivalence. Suppose there is more than one value for the ordered sequence (a, b, c,...) over GF (q), say (a 1, b 1, c 1,...) and (a 2, b 2, c 2,...) are two distinct ordered sequences with corresponding matrices A 1 = I r D 1 and A 2 = I r D 2. Then A 1 and A 2 are projective inequivalent representations of M. However, if there is a linear transformation followed by a column permutation that maps A 1 to A 2, then A 1 and A 2 are geometric equivalent. Otherwise they are geometric ineqivalent representations of M. Geometric equivalence is an equivalence relation on the set of sequences (a, b, c,...) over GF (q). Two sequences are geometric equivalent if they belong to the same equivalence class. The number of geometric inequivalent representations is the number of equivalence classes, and if there is only one equivalence class, then M is uniquely representable over GF (q) with respect to geometric equivalence. By construction the number of geometric inequivalent representations is at most the number of projective inequivalent representations. To complete the proof we must show that the computational complexity of determining if two projective inequivalent representations are geometrically inequivalent is polynomial in n. This is done in the next lemma. Lemma 2.2. Let A and B be two n-element rank-r matrices over GF (q) with r n. Then there exists an algorithm of complexity no more than q r2 rn 2 for determining whether a non-singular matrix R M r r (GF (q)) and a permutation matrix C M n n (GF (q)) exist such that R A C = B. Proof. Let V (r, q) denote the r-dimensional vector space over GF (q). We want to determine whether B can be obtained from A by applying a row-wise linear transformation over V (r, q), followed by a column permutation. That is, we wish to determine whether there is a nonsingular matrix R M r r (GF (q)) and a permutation matrix C M n n (GF (q)) such that R A C = B. Given a candidate matrix R, the complexity of checking whether a corresponding C exists is rn 2, as we simply need to check that R A and B have the same columns. Thus, the complexity of the entire algorithm is xrn 2, where x is the complexity of enumerating the r r nonsingular matrices over GF (q); that is, the elements of the general linear group GL(r, GF (q)). In the worst case this portion is O(q r2 ), if every r r matrix over GF (q) must be tested. Therefore, the complexity of the entire algorithm is no more than q r2 rn 2. Let us illustrate the theorem with an example. The matroid Q 6 is shown in Figure 3. The fundamental circuits corresponding to the basis {1, 2, 3} are {1, 2, 4}, {2, 3, 5}, and {1, 2, 3, 6}. The matrix obtained from the fundamental circuits is I The corresponding bipartite graph is shown in Figure 3 with a spanning tree highlighted. The entries corresponding to the edges of the spanning tree may be taken as 1 and the remaining entries are denoted by (a, b) where a and b are non-zero members of the field. 101 Copyright SIAM.

5 Figure 3: Coordinatizing Q 6 Thus we can write the matrix representing Q 6 as I a 0 1 b B 4 and B 6 and matrix B 3 is geometric equivalent to B 5. An exhaustive search found no mapping from B 1 to B 3. Thus, we conclude that Q 6 has two geometric inequivalent representations over GF (5), B 1 and B 3. To confirm our exhaustive search we calculated the single-element extensions of B 1 and B 3. The matroid MB 1 has nine non-isomorphic simple singleelement extensions over GF (5), whereas the matroid MB 3 has seven non-isomorphic simple singleelement extensions. The first six matroids in Figure 4 are common single-element extensions of MB 1 and MB 3. The next three matroids are single-element extensions of MB 1, but not of MB 3. The last matroid is a single-element extension of MB 3, but not of MB 1. Observe that Over GF (3) the choices for the ordered sequence (a, b) are (1, 1), (1, 2), (2, 1) and (2, 2). The first gives W 3 and the rest give W 3. This implies that Q 6 is not representable over GF (3). Over GF (5) the ordered sequence (a, b) may be (1, 1), (1, 2), (2, 1), (1, 3), (3, 1), (1, 4), (4, 1), (2, 2), (2, 3), (3, 2), (2, 4), (4, 2), (3, 3), (3, 4), (4, 3), or (4, 4). Of these (1, 1) gives W 3 ; (1, 2), (2, 1), (1, 3), (1, 4), (2, 2), (3, 2), (3, 3), (4, 3), and (4, 4) give W 3 and the remaining give Q 6. Let us refer to the matrices with (a, b) equal to (3, 1), (4, 1), (2, 3), (2, 4), (4, 2) and (3, 4) as B 1, B 2, B 3, B 4, B 5, and B 6, respectively. This implies that Q 6 has six projective inequivalent representations over GF (5) The six projective inequivalent representations are shown below: B 1 = I B 2 = I B 3 = B 5 = I I B 4 = B 6 = I I Matrices B 1 and B 2 are geometric equivalent because B 1 can be transformed to B 2 as follows: multiply row 2 by 4; swap rows 1 and 3; replace row 2 by row 2 + row 1 + row 3 ; multiply column 2 by 4; swap columns 1 and 5 and columns 3 and 4. Similarly, we can check that matrix B 1 is geometric equivalent to Figure 4: Single-element extensions of Q 6 over GF (5) As another example, we can check that P 7 (from 102 Copyright SIAM.

6 Figure 2) has three projective inequivalent representations over GF (5), but is uniquely representable with respect to geometric equivalence. Whereas, over GF (7) it has two geometric inequivalent representations. When the concept of projective equivalence was developed in 9 there were no examples of geometric inequivalent representations. As we see in this paper, it is too tedious to compute by hand even for small matroids. However, with the subsequent use of computers in matroid theory research, now there is no shortage of examples and computationally it doesn t cost much to determine which of the projective inequivalent representations are also geometric inequivalent. Finally, note that if A 1 and A 2 have different isomorphism classes of single-element extensions over GF (q), then A 1 and A 2 are geometric inequivalent representations. However, the converse is not true. For example, W 3 over GF (7) has three geometric inequivalent representations, but two of them have exactly the same single-element extensions. 3 Computing single-element extensions with geometric equivalence In this section we present a method for generating single-element extensions with respect to geometric equivalence. We illustrate the method by computing the single-element extensions of the relaxed Fano matroid, F7, over GF (5). This same example is presented in 4 as an example of how to compute projective inequivalent extensions. Although not necessary for understanding this paper, the reader may find it interesting to compare and contrast 4 on projective equivalence with this paper on geometric equivalence, to see how the computational perspective naturally leads to the development of the concept of geometric equivalence. Let us denote the single-element extension of a matroid M by an element e as M + e 7, p Comparing the matrix representing F7 (shown below) to the matrix representing P G(2, 5) and adding to A the columns in P G(2, 5) missing from A gives three isomorphism classes. Each class contains several choices for the new column representing the element e in A + e : A = 1 I Class 1 {{0, 1, 2, 0, 1, 3}, {1, 2, 0, 1, 3, 0}, {1, 0, 2, 1, 0, 3}, {1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 3, 1, 1, 4, 1, 1, 4, 4}} Class 2 {{0, 1, 4}, {1, 4, 0}, {1, 0, 4}, {1, 1, 2, 1, 2, 1, 1, 3, 3}} Class 3 {{1, 2, 3, 1, 2, 4, 1, 3, 2, 1, 3, 4, 1, 4, 2, 1, 4, 3}} The three matroids corresponding to the three isomorphism classes, M 1, M 2, and M 3 are shown in Figure 5. The new element is circled in each matroid. Figure 5: Single-element extensions of F 7 over GF (5) Within each isomorphism class, group together the columns with non-zero entries in the same position and check if the resulting matroids are equal. In the first isomorphism class there are four such groups. In the first group, the matroid obtained by adding column 0, 1, 2 to A is equal to the matroid obtained by adding column 0, 1, 3 to A. However, A with column 0, 1, 2 can be transformed into A with column 0, 1, 3 as follows: swap rows 2 and 3; multiply rows 2 and 3 by 4; replace row 2 by row 2 +row 1 ; replace row 3 by row 3 + row 1 ; multiply columns 2, 3, and 5 by 4 and column 8 by 2; and swap columns 1 and 7. Thus these two representations are projective inequivalent, but geometric equivalent. Similarly the matroid obtained by adding column 1, 2, 0 to A is equal to the matroid obtained by adding column 1, 3, 0 to A, and we can check that the corresponding matrices are geometric equivalent. The matroid obtained by adding column 1, 0, 2 to A is equal to the matroid obtained by adding column 1, 0, 3 to A and the two matrices are geometric equivalent. This takes care of the first three groups in Class 1. The situation is a bit more complicated in the last group containing columns {1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 3, 1, 1, 4, 1, 1, 4, 4}} Here the matroid obtained by adding column 1, 1, 3 to A is equal to the matroid obtained by adding column 1, 1, 4 to A. The matroid obtained by adding column 1, 2, 2 to A is equal to the matroid obtained by adding column 1, 4, 4 to A. The matroid obtained by adding column 1, 3, 1 to A is equal to the matroid obtained by adding column 1, 4, 1 to A. But the three matroids, A with columns 1, 1, 3, 1, 2, 2, and 1, 3, 1, respectively, are not equal (see Figure 6). We can check that in each of the three cases the corresponding matrices are geometric equivalent. Thus we may conclude that F7 extends to M 1 by giving rise to two projective inequivalent represen- 103 Copyright SIAM.

7 tations of M 1. However, F7 extends to M 1 uniquely with respect to geometric equivalence. Figure 6: Drawings of the first isomorphism class of F7 over GF (5) For the second isomorphism class, there are three groups with one column each and a fourth group with three columns. The three matroids in the fourth group, A with columns 1, 1, 2, 1, 2, 1, and 1, 3, 3, respectively, are not equal. So F7 extends to M 2 without giving rise to any projective inequivalent extensions of M 2 and consequently without giving rise to geometric inequivalent representations of M 2. For the third isomorphism class, there is just one group of columns. The matroid obtained by adding column 1, 2, 3 to A is equal to the matroid obtained by adding column 1, 3, 4 to A, and the corresponding matrices are geometric equivalent. The matroid obtained by adding column 1, 2, 4 to A is equal to the matroid obtained by adding column 1, 4, 2 to A, and the corresponding matrices are geometric equivalent. The matroid obtained by adding column 1, 3, 2 to A is equal to the matroid obtained by adding column 1, 4, 3 to A, and the corresponding matrices are geometric equivalent. The three matroids, A with columns 1, 2, 3, 1, 2, 4, and 1, 3, 2, respectively, are not equal. So, F7 extends to M 3 by giving rise to two projective inequivalent representations of M 3. However, we can check that it extends to M 3, uniquely with respect to geometric equivalence. Theorem 3.1. Let M be a connected simple matroid represented by A = I r D over GF (q) and let P = I r L represent the projective geometry P G(r 1, q). Let A 1 = I r D x 1 and A 2 = I r D x 2 be matrices representing the same single-element extension M +e where x 1 and x 2 are distinct columns in L D with non-zero entries in the same position. Then (i) A 1 and A 2 are projective inequivalent representations of M + e if and only if MA 1 = MA 2. (ii) A 1 and A 2 are geometric inequivalent representations of M + e if and only if MA 1 = MA 2 and there is no non-singular linear transformation followed by a column permutation from A 1 to A 2. Proof. A representation of P G(r 1, q) in standard form may be written as P = I r L by selecting from each one-dimensional subspace of V (r, q), the column with one in the first non-zero position. If I r D is a matrix over GF (q) representing M, then I r D x represents M +e, where x is the column in the matrix P corresponding to element e. Observe that there may be several such columns all representing the same element e. For each single-element extension M + e, let X be a set of columns in L D such that for every x X, I r D x is an GF (q)-representation of M + e. In other words X is the set of columns that give the same single-element extension up to isomorphism. We can group the columns in X as follows: x 1 and x 2 are in the same group if they have non-zero entries in the same positions. Let A 1 = I r D x 1 and A 2 = I r D x 2, where x 1 and x 2 are columns from the same group. Then A 1 and A 2 are projective inequivalent representations if and only if MA 1 = MA 2. However, if a linear transformation followed by a column permutation can be found between A 1 and A 2, then by definition they are geometric equivalent. The proof of the next result follows immediately from Theorems 2.1 and 3.1. Corollary 3.2. Let M be an n-element rank r GF (q)-representable matroid. The set of distinct matrix representations of M whose singleelement extensions must be computed to obtain all the non-isomorphic single-element extensions of M over GF (q) can be obtained from the set of projective inequivalent representations using a polynomial time algorithm in n. We will end this paper with some comparisions between projective and geometric inequivalence. Since a matroid can have a large number of projective inequivalent representations, the concept of stability was developed as a way of bounding this number. A minor N of a matroid M stabilizes M over GF (q) if no GF (q)-representation of N can be extended to two projective inequivalent GF (q)-representations of M. Observe that if N has k projective inequivalent representations, then M has at most k projective inequivalent representations. In the previous example, F7 stabilizes its second extension, but does not stabilize its first and third extension. There is no analog of the concept of stability for geometric equivalence. For example, as mentioned earlier, W 3 has two 104 Copyright SIAM.

8 geometric inequivalent representations over GF (5). One of its single-element B 7 (see figure 2) is uniquely representable with respect to geometric equivalence. However, a single-element extension of B 7 has two geometric inequivalent representations over GF (5). With geometric equivalence we are able to ask questions such as, given a fixed finite field GF (q) and a fixed rank r, is there an n beyond which the matroids become uniquely representable with respect to geometric equivalence? We know the projective geometry is uniquely representable with respect to equivalence and hence also with respect to geometric equivalence. It is very likely that there is an early threshold for geometric equivalence after which all the rank-r GF (q)-representable matroids are uniquely representable. References 1. T. H. Brylawski and D. Lucas (1976). Uniquely representable combinatorial geometries. In Teorie combinatorie (Proc Internat. Colloq.), Accademia Nazionale dei Lincei, Rome. 2. J. Geelen, G. Whittle (2010). The projective plane is a stabilizer, J. of Combin. Theory Ser. B, 100, no P. Hliněný (2006). Equivalence-free exhaustive generation of matroid representations. Discrete Applied Mathematics, 154, S. R. Kingan (2009). A computational approach to inequivalence in matroids. Proceedings of the Thirty-Seventh Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 198, S. R. Kingan (2011). Restrictions as Stabilizers, Advances in Applied Mathematics, 47, J. Kahn, (1988). On the uniqueness of matroid representations over GF (4), Bull. London Math. Soc. 20, J. G. Oxley (1992). Matroid theory, (1992), Oxford University Press, New York (second edition, 2011). 8. J. G. Oxley, D. Vertigan, and G. Whittle (1996). On inequivalent representations of matroids over finite fields. Journal of Combinatorial Theory, Series B, 67, G. Whittle (1999). Stabilizers of classes of representable matroids, J. Combin. Theory Ser. B Copyright SIAM.