Theory of matroids and euclidean representation

Question

I am looking for the definition of the "euclidian representation" of a matroid. I guess it is used for graphic matroids but impossible to find the construction for this representation. Thank you for your help.

Answer 1

sorry for my english because I am French. I can answer your question in the case of a matroid of rank 3 (I don't know in different rank, but I think you are interested in this case). Indeed in the case of a graphic matroid M, a theorem claims that M is euclidean and so you are sure to have an euclidean representation. You construct it as follow :

• let E be the edges of M with #E=k (and V its vertices)

• you call e(1), ..., e(k) those edges

• at each edge e(i) in your matroid M corresponds a vertex in your euclidean representation
• then you link three points in the euclidean representation iff the three associated edges form a loop in M

Then, the rule to interpret and euclidean representation is that 3 vertices in the representation form a base in M iff they are not linked in the representation (remember that M is of rank 3)

Your description is complete because :

• a set of 3 points is a loop iff it is linked in the representation
• a set of 3 points is a base iff those 3 points are not in a same line
• a set of more than 3 points is a loop iff if you delete points you don't obtain three points linked by a line (because it would says you that there is a loop in this set of points and then it is not minimal)

I hope I have answered your question (not easy without a drawing ...)

Answer 2

Let $M$ be a matrix over a field $\mathbb{F}$. It is easy to show that the set $\mathcal{I}$ consisting of all linearly independent subsets of columns of $M$ form the independent sets of a matroid $\mathcal{M}$. This matroid is called the vector matroid of the matrix $M$ over $\mathbb{F}$.

An arbitrary matroid $\mathcal{M}$ is representable (a.k.a. realizable) over a field $\mathbb{F}$ if it is the vector matroid of some matrix with entries in $\mathbb{F}$. If $\mathbb{F} = \mathbb{E}$ is euclidean space and $\mathcal{M}$ is representable over $\mathbb{E}$, then any matrix $M$ whose vector matroid is $\mathcal{M}$ (there are many of them) is a euclidean representation of $\mathcal{M}$.