1. Introduction

Matroid theory dates from the 1930’s when Whitney first used the term matroid in his basic paper [1] . Matroid theory is a common generalization of linear independence of graphs and matrices. One of the mainly interesting things of matroid theory is the representability of a matroid. One problem of representability is to find the fields over which the given matroid is representable. The other problem is to find the excluded minors for which the matroid is representable over the given field. It was found the complete set of excluded minors for two or three element fields [2] - [5] . In 1984, Kahn and Seymour conjectured that is the complete set of ex- cluded minors for GF(4)-representability. Oxley showed that the conjecture is wrong by showing is also excluded minor for GF(4)-representability in his brief note [6] . Geelen, Gerarads and Kapoor proved that it is enough to add to the list of Kahn and Seymour [7] . It is not an easy problem to find the excluded minors for GF(q)- representability when q is more than 4. Instead, we have the conjecture by Rota that the number of excluded minors are finite for any prime powers q [8] . In this paper, we study the properties of minor and show that is excluded minor for GF(4)- representability deliberately.

2. Preliminaries

2.1. Matroid

Matroid theory has exactly the same relationship to linear algebra as does point set topology to the theory of real numbers. That is, point set topology postulate the pro- perties of the open sets of real line and matroid axiomatize the character of the in- dependent set in vector space:

Definition 2.1. A matroid M is a finite set E and a collection of subsets of E satisfying the following three conditions:

(I₁).

(I₂) If and then.

(I₃) If X, Y are in and, then there exists such that .

(C₁).

(C₂) If and are in and, then.

(C₃) If and are distinct members of and, then there is member of such that.

Now, let be a subset of the power set of a finite set E. If satisfies the conditions (C₁), (C₂) and (C₃), then is called the set of circuits of a matroid on E. Let be the set of circuits of a matroid M. Then, the set of all subsets of E which contain no member of satisfies the independent conditions (I₁), (I₂) and (I₃). Also for a matroid, the set of all minimal dependent set M satisfies the three circuits conditions. Thus, the matroid defined by circuits is the same as the one defined by independent sets. We need two other definitions of a matroid.

Definition 2.2. Let E be a finite set and r be a function from to the set of non- negative integers and satisfies the following conditions:

(R₁) If, then.

(R₂), then.

(R₃) If X and Y are subsets of E, then. Then, r is called the rank function of a matroid M on E. Let M be the matroid and suppose that. Let be. Then, it is easy to see that the pair is a matroid. We call this matroid the restriction of M to X or the deletion of from M. It is denoted by or. We define the rank of X to be the size of a maximal independent set of. This function which is called the rank function of M satisfies the conditions (R₁), (R₂) and (R₃). This function is denoted by and will be denoted by. On the other hand, if r is a rank function of a matroid M, the set is the set of in- dependent set of M. Thus, the definition of a matroid by a rank function is equivalent to the definition by independent sets. One of the definitions of matroid is the one by closure operator. Throughout this thesis, by E means a finite set.

Definition 2.3. Let cl be a function from to satisfying the following;

(CL₁) If, then.

(CL₂) If, then.

(CL₃) If, then.

(CL₄) If, , and, then.

Then cl is called the closure operator of a matroid M on E. Let M be a matroid on E with the rank function r. Define cl to be the function from to by

for all. Then, we can see that cl satisfies (CL₁)-(CL₄). On the other hand, if cl is the closure operator of a matroid M on E, then satisfies the independent axioms and is the matroid having closure operator cl. Thus, the four definitions of matroid defined by the above are equivalent. We can see that there are a lot of equivalent definition of matroid. We have to introduce one more definition of matroid. For a matroid M, the maximal independent set in M is called a basis or base of M. It is easy to see that every basis element has the same cardinal number. Let be the set of all base element of M. Then, has the properties;

(B₁) is non-empty.

(B₂) If and are in and then there is such that.

Conversely, let be a collection of subsets of E satisfying the axioms (B₁) and (B₂). And let. Then is a matroid having as its collection of bases. If M is a matroid and, then we call the closure or span of X in M, and we write this as. If, then X is called a flat or a closed set of M. A hyperplane of M is a flat of rank. A subset X of is a spanning set of M if. Let M be a matroid and be. Then, satisfies the following axiom which is equivalent to (B₂):

(B₂)* If and are in and, then there is an element such that. The matroid having the set of all basis element B^*(M) is called the dual of M. Thus and. Also. The bases of are called cobases of M. Similarly, the circuits, hyperplanes, independent sets and spanning sets of are called cocircuits, cohyperplanes, coindependent sets, and cospanning sets of M. The next result gives some elementary relationships between these sets.

Proposition 2.4. Let M be a matroid on a set E and suppose. Then

1) X is independent if and only if is cospanning.

2) X is spanning if and only if is coindependent.

3) X is a hyperplane if and only if is cocircuit.

4) X is a circuit if and only if is a cohyperplane.

Proof. 1) Let X be an independent set in M. Then, for some basis element B of M. Thus and, where is the clo- sure in. If is cospanning, then for some basis element of. It means that and X is independent set. 2) is deduced by applying 1) to. 3) is obtained by the following equivalent statements; a) X is a hyperplane of M. b) X is a non-spanning set of M but is spanning for all. c) is dependent in but is independent in for all. d) is a cocircuit of M. 4) is the one obtained by applying 3) to.

□

Let’s remind of the finite fields. If F is a finite field, then F has exactly p^k-elements for some prime p and some positive integer k. Indeed, for all such p and k, there is a unique field having p^k-elements. This field is called the Galois field of order p^k. When, coincides with, the ring of integers modulo p. When, can be constructed as follows. Let be a polynomial of degree k with coefficients in and suppose that the polynomial is irreducible. Consider the set S of all polynomials in that have degree at most and have coefficients in. There are exactly p choices for each of the k-coefficient of a member of S. Hence,. If we take p = 2 and k = 2 we get the field GF(4). Moreover, under addition and multiplication, both of which are performed modulo, S forms a field namely. In case of, we can take the irreducible polynomal to .

Let A be a matrix over a field F. Then, the collection of independent column vectors of A satisfies the independent axioms of matroid. Thus, is a matroid and this matroid is denoted by, where is the set of all column vectors of A.

Now, let G be a graph. Then is the matroid on the edge set with the set of all cycles of G as circuit. This matroid is called the cycle matroid of G. Two matroids and are isomorphic, denoted by, if there is a bijection from to such that, for all, is independent in if and only if X is independent in. A matroid that is isomorphic to the cycle matroid of a graph is called graphic. If M is isomorphic to for a matrix A over F, then M is called F-representable. In the sequel, by F we mean a finite field.

We call an element e a loop of a matroid M if is a circuit of M. Moreover, if f and g are element of such that is a circuit, then f and g are said to be parallel in M. A parallel class of M is a maximal subset X of such that any two distinct members X are parallel and no member of X is a loop. A parallel class is trivial if it contains just one element. If M has no loops and no non-trivial parallel classes, it is called a simple matroid or a combinatorial geometry.

2.2. Uniform Matroid U_m,n

Let m and n be non-negative integers such that. Let E be a set of cardinality n and be all subsets of E of cardinality less than or equal to m. This is a matroid on E, called the uniform matroid of rank m and denoted by. By definition, the set of basis of is, and the set of circuits is.

2.3. Affine Matroid

Now, we are going to define affine matroid. A set is affinely dependent if and there are elements of F, not all zero, such that

and. It is easy to show that affine dependence of

is equivalent to each of the followings;

(Ad₁) is linearly dependent, where is the - tuple of elements of F.

(Ad₂) is linearly dependent.

A set is affinely independent if it is not affinely dependent.

Suppose that. Let A be the matrix over F, the i-th column of with is. The matroid is called an affine matroid over F. In particular, if, and, then the affine matroid is denoted by. In general, if M is an affine matroid over of rank, where, then a subset X of is dependent in M if, in the representation of X by points in, there are two identical points, or three collinear points, or four coplanar points, or five points in space. Hence the flats of M of ranks one, two, and three are represented geometrically by points, lines, and planar, respectively.

We extend the use of diagram of affine matroid to represent arbitrary matroids of rank at most four. Generally, such diagrams are governed by the following rules. All loops are marked in a single inset. Parallel elements are represented by touching points. If three elements form a circuit, the corresponding points are collinear. Likewise, if four elements form a circuit, the corresponding points are coplanar. In such a diagram, the lines need not be straight and the planes may be twisted. Certain lines with fewer than three points on them will be marked as part of the indication of a plane, or as con- struction lines. We call such a diagram a geometric representation for the matroid.

Now we will define the projective geometry. Let V be a vector space over F. For each, if v and w lie on the same 1-dimensional subspace of V. Then, ~ is an equivalence relation on V and is called the projective space of V or projective geometry and will be denoted by. For a matroid M, delete all the loops from M and then, for each non-trivial parallel class X, delete all but one distinguished element of X, the matroid we obtain is called the simple matroid associated with M and is denoted by. Evidently the construction of from V is analoguous to the construction of the simple matroid from a matroid M. It is clear that a matroid M is F-representable if and only if its associated simple matriod is F-representable. Hence, when we discuss representability questions, it is enough to concentrate on simple matroids.

2.4. Projective Geometrices

If, then has dimension n and it is denoted by. In particular, when F is, it will be written for. Let’s find the geometric representation of. For each, there are no non- zero elements except v on the 1-dimensional subspace of through v (Figure 1). Thus

.

It is easy to see and lie on the same line (plane) of (). Also, and are circuits of. Furthermore, is a circuit, because. Thus the geo- metric representation of is Figure 2.

is called the Fano matroid and will be denoted by. In, is a circuit and a hyperplane. The matroid N obtained from by relaxing the circuit hyperplane is called non-Fano matroid and is denoted by (Figure 3).

2.5. Duals of Representable Matroids

Give a matrix A by elementary row operations and interchanging two columns

or deleting a zero row. A can be transformed to a form, where is identity matrix and D is matrix. It is clear that is isomorphic to. We can show that the dual matroid of M is ( [9] ). Hence, the dual of F-representable matroid is F-representable. For example, let

be a matrix over. Then, we can see that is isomorphic to.

and we can see that the set of circuits of is

.

3. Minors

In this section, we define minors which are important to representability. For the definition of minor, we have to define contraction which is the dual of the operation of deletion. We can see that contraction for matroids generalizes the operation of contraction for graphs. Let M be a matroid on E and T be a subset of E. Then is called the contraction of T from M and also denoted by. For easy understanding, let us see what it means in graphic matroids. Let G be a graph

and. Then, G/3 is the graph

and is

Also is

and is

which is the same as G/3. Thus,

and we see that the contraction of a graphic matroid is the same as the matroid of the contracted graph, where we used for a planar graph G.

Now let be the dual of a matroid M. Then the rank function of is given by ( [9] ). If, the rank function of is the restriction of to the subset of, that is, for all, .

Proposition 3.1. If, then for all,.

Proof. By definition,. Thus

because and. □

Proposition 3.2. Let be a basis for. Then

Proof. For the convenience, let’s denote the equality by. It is clear that. To show that, let. Then, for some basis B of. Cleary is a basis of. So. Thus

and it was proved that. Now if we show that, the proof is completed. Let. Then

since is a basis of. Hence and. □

Proposition 3.3. If, then

1),

2), and

3).

Proof. 1). Thus.

2). 3) is obtained if we replace M by in the left-

hand side of 1). □

Now, let A be a matroid over F and T be a subset of the set E of column levels of A. We shall denote by the matrix obtained from A by deleting all the columns whose labels are in T. Clearly,. Moreover, by the following, we can see that the class of F-representable matroids is minor closed.

Proposition 3.4. Every contraction of an F-representable matroid is F-representable.

Proof. The duals of F-representable matroid are F-representable. Since

, we proved that a contraction of F-representable matroid is F-representable. □

Now suppose that e is the label of a non-zero column of A. Then, by pivoting on a non-zero entry of e, we can transform A into a matrix in which the column labelled by e has single non-zero entry. In this case, will denote the matrix obtained from by deleting the row and column containing the unique non-zero entry in e. Then, we have the following property.

Proposition 3.5..

Proof. It is enough to show that the second equation is true, because the first equation is clear. By using row and column swaps if necessary, can be considered as the matrix in which the unique nonzero entry of e is in row 1 and column 1. Let I be a k-element subset of the ground set of such that. Then the set of columns labelled by is linearly independent if and only if the matrix B which has columns and the 1st column of it is the column corresponding to e has rank. This is equivalent to the matrix deleted row 1 and column 1 of B has rank k and this is equivalent to the columns of labelled by I is linearly independent. Thus,.

□

4. Representability of P₈

Now, we shall describe the construction of representations for matroids. Two matrices and are equivalent if and are isomorphic. It is easy to see that if is a rank-r matroid, then A is equivalent to a standard matrix, where is the identity matrix. Given such a matrix, let its columns be labelled in order,. Let B be the basis of. For all i in, the unique non-zero entry in column i of is in row i. Thus it is natural to label the rows of by. Hence, D has its rows labelled by and its columns labelled by. For all, there exists a unique circuit contained in. In fact,

. is called the B-fundamental circuit of. Let be the matrix obtained from D by replacing each non-zero entry of D by 1. Then the columns of are precisely the incidence vectors of the sets. This matrix is called the B-fundamental-circuit incidence matrix of. Now let be a rank-r matroid and B be a basis for M. Let X be the B -fundamental-circuit incidence matrix of M. And let columns of X be labelled by. Then,. Thus the task of finding an F-representation for M can be viewed as being one of finding the specific elements of F that correspond to the non-zero elements of. We can see that most of the entries of D can be predetermined by the following Proposition 4.1. Before stating it, we shall require some preliminaries.

Let the rows of be indexed by and its columns by. Let denote the associated simple bipartite graph, that is, has vertex classes and and two vertices and are ad- jacent if and only if the entry in row and column of is 1. For example, if

then is

We have a nice way for representing matroid;

Proposition 4.1. Let the matrix be an F-representation for the matroid M. Let be a basis of the cycle matroid of. Then, where is the number of connected component of. Moreover, if is an ordered k-tuple of non-zero elements of F, then M has an F-representation that is equivalent to such that, for each i in, where the entry of corresponding to is ( [9] [10] ).

By the above proposition, we can find the fields on which given matroid is re- presentable.

Example 4.2. is the matroid whose geometric representation is the following Figure 4.

Let be a representation over of. Then

Thus, the associated simple bipartite graph is

and a basis of is

Therefore, is equivalent to the form of

, , ,

, ,.

We want to find the fields in which the negative equations satisfy. It is easy to see that the equations have no solution if is equal to or. In case of, let’s check if the equations have a common solution. There are three cases;

1).

If, then. Thus, and. Hence, we don’t have a solution.

2).

In this case, and. Thus, there are no solution.

3).

In this case, and. Thus we have no solution.

In, if, then and . Therefore we showed that is representable over F if and only if.

Example 4.3. If

is the matrix over F with, then it is easy to see that by Figure 3. Also, if is a matrix over F and, then

By taking a basis of, is equivalent to a matrix

Since. Also, because and ,. Thus.

Since,. Therefore, we showed that is repre- sentable over F if and only if. is the matroid of the matrix

over. It’s geometric representation is the following Figure 5.

Lemma 4.4. is representable over a field F if and only if.

Proof. If, then the B-fundamental circuit incidence matrix of is

By taking a basis of, A is equivalent to a matrix, where

Because and and.

As. Therefore

where. Therefore, we proved that is representable over F if and only if. □

Non F-representable matroid for which every proper minor is F-representable is called the excluded or forbidden minor for F-representability. Because a matroid M is F-representable if and only if all its minors are F-representable (Proposition 3.5), finding the complete set of excluded minors for F-representability is the solution for the F-representability. Since the duals of F-representable matroid are F-representable, the dual of an excluded minor for F-representability is an excluded minor for F-representability.

To find an excluded minor for -representability, we need the following property:

Proposition 4.5. Let F be a field and k be an integer exceeding 1. Then uniform matroid is F-representable if and only if.

Proof. Let, where A is a matrix. We can consider A as a matrix

where are non-zero different elements of F. Thus, it should be and so. Conversely, if, then and we can choose non-zero different -elements of F.

□

From the above proposition, we can see that and are excluded minors for GF(q)-representability. In 1958, Tutte showed that is the only excluded minor for -representability ( [2] ). The problem of finding the com- plete set of excluded minors for -representability was solved by Bixby and Seymour in 1979 ( [3] [4] ). The set is.

By Proposition 4.5, it is easy to see that and are excluded minors for GF(4)-representability. In Examples 4.2 and 4.3, we can see that and are not GF(4)-representable. It is not difficult to see that every proper minor of them is GF(4)-representable. Thus and are excluded minors for GF(4)-represent- ability. Clearly is self-dual.

Let

be the matrix over. Then, is the matroid.

Lemma 4.6. is representable over a field F if and only if.

Proof. Let be an F-representation for. If, then the B- fundamental circuit incidence matrix for is

By choosing a basis for, we can consider

Because and and. Substituting these to the matrix D, we have

From the circuits, , and, we get the equations

From the first and fourth equation, we get. Substituting b for d in the second and third equation, we get and. As, it follows that and. Because,. Thus

In fact, we can show that □

For a matroid M, an automorphism is a permutation of such that for all. The set of automorphisms of M forms a group under composition. This automorphism is transitive if, for every two elements x and y of, there is an automorphism that maps x to y.

Lemma 4.7. The automorphism group of acts transitively on.

Proof. We can see that the geometric representation of is the following Figure 6

because has only 10 4-circuits and. From the geometric representation of, it is easy to see that the permutations and are both automorphisms of. For example, and are the following Figure 7 and Figure 8.

Thus the automorphism group of is.

Any two elements in and can be mapped to each other by an automorphism in. Similarly, any two elements in and can be mapped to each other by an automorphism in. For the remaining two elements of, they are mapped each other by the following;, , , ,

, , , .

Thus, the automorphism group of acts transitively on.

□

Now, we get the following result, which is the purpose of this paper.

Theorem 4.8. is an excluded minor for GF(4)-representability.

Proof. By Lemma 4.6, is not representable. Because the automorphism group of acts transitively on by Lemma 4.7, for any element e of, we have.

By Proposition 3.5,. Because

. But, since is representable over by Lemma 4.4, every contraction of is GF(4)-representable. By Proposition 3.3(1), for each element,. Because is self-dual,. Thus

. Hence, every deletion of is GF(4)-representable. Therefore, every proper minor of is GF(4)-representable and we proved that is an excluded minor for GF(4)-representability. □