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One of the mainly interesting things of matroid theory is the representability of a matroid. Finding the set of all excluded minors for the representability is the solution of the representability. In 2000, Geelen, Gerards and Kapoor proved that

Matroid theory dates from the 1930’s when Whitney first used the term matroid in his basic paper [

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

(I_{1})

(I_{2}) If

(I_{3}) If X, Y are in

By the definition, for a finite vector space V and for the collection of linearly independent subsets

(C_{1})

(C_{2}) If

(C_{3}) If

Now, let _{1}), (C_{2}) and (C_{3}), then _{1}), (I_{2}) and (I_{3}). Also for a matroid

Definition 2.2. Let E be a finite set and r be a function from

(R_{1}) If

(R_{2})

(R_{3}) If X and Y are subsets of E, then_{1}), (R_{2}) and (R_{3}). This function is denoted by

Definition 2.3. Let cl be a function from

(CL_{1}) If

(CL_{2}) If

(CL_{3}) If

(CL_{4}) If

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

_{1})-(CL_{4}). On the other hand, if cl is the closure operator of a matroid M on E, then

(B_{1})

(B_{2}) If

Conversely, let _{1}) and (B_{2}). And let_{2}):

(B_{2})* If ^{*}(M) is called the dual of M. Thus

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

1) X is independent if and only if

2) X is spanning if and only if

3) X is a hyperplane if and only if

4) X is a circuit if and only if

Proof. 1) Let X be an independent set in M. Then,

□

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 ^{k}-elements. This field is called the Galois field of order p^{k}. When

Let A be a matrix over a field F. Then, the collection of independent column vectors

Now, let G be a graph. Then

We call an element e a loop of a matroid M if

Let m and n be non-negative integers such that

Now, we are going to define affine matroid. A set

(Ad_{1})

(Ad_{2})

A set

Suppose that

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

It is easy to see

Give a

or deleting a zero row. A can be transformed to a form

be a matrix over

and we can see that the set of circuits of

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

and

and

Also

and

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

Now let

Proposition 3.1. If

Proof. By definition,

because

Proposition 3.2. Let

Proof. For the convenience, let’s denote the equality by

and it was proved that

since

Proposition 3.3. If

1)

2)

3)

Proof. 1)

2)

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

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

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

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

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,

□

Now, we shall describe the construction of representations for matroids. Two matrices

Let the rows of

then

We have a nice way for representing matroid;

Proposition 4.1. Let the

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

Example 4.2.

Let

Thus, the associated simple bipartite graph

and a basis of

Therefore,

We want to find the fields

1)

If

2)

In this case,

3)

In this case,

In

Example 4.3. If

is the matrix over F with

By taking a basis of

Since

Since

over

Lemma 4.4.

Proof. If

By taking a basis of

Because

As

where

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

Proposition 4.5. Let F be a field and k be an integer exceeding 1. Then uniform matroid

Proof. Let

where

□

From the above proposition, we can see that

By Proposition 4.5, it is easy to see that

Let

be the matrix over

Lemma 4.6.

Proof. Let

By choosing a basis for

Because

From the circuits

From the first and fourth equation, we get

In fact, we can show that

For a matroid M, an automorphism is a permutation

Lemma 4.7. The automorphism group of

Proof. We can see that the geometric representation of

because

Thus the automorphism group of

Any two elements in

Thus, the automorphism group of

□

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

Theorem 4.8.

Proof. By Lemma 4.6,

By Proposition 3.5,

Ahn, S. and Han, B. (2016) Excluded Minority of P_{8} for GF(4)- Representability. Open Journal of Discrete Mathematics, 6, 279-296. http://dx.doi.org/10.4236/ojdm.2016.64024