Sorry if this question is a dumb one.

I used a special family of matroids in my research. One of them, of rank 3, can be represented by the following matrix over $\mathbb F_5$ or over $\mathbb R$:

$\left(\begin{array}{cccccccccccc} 1&0&0&1 &1 &0 &-1&2 &-1& 2&0 &0\\ 0&1&0&-1&0 &1 &2 &-1&0 & 0&-1&2\\ 0&0&1&0 &-1&-1&0 &0 &2 &-1&2 &-1 \end{array}\right) $

It contains 12 points, 3 lines of 5 points, 7 lines of 3 points, all the other lines are of 2 points. It is a restriction of the Dowling geometry $Q_3(\mathbb F_5^\times)$.

The other matroids of this type are represented over $\mathbb R$ by the matrix, the columns of which are all possible column vectors of length $n$ (the parameter) which contain either one nonzero entry (equal to $1$) or two such entries (equal to $1$ and $-1$, or $2$ and $-1$).

I would like to ask: is this class of matroids known? Any reference is welcome.

Here is a geometric representation of the matroid above. The picture should be seen as a figure in projective plane, the dashed circle corresponding to the line at infinity.



1 Answer 1


Matrices that contain at most two non-zero entries per column are called frame matrices. The matroids representable by frame matrices (over a finite field $\mathbb{F}$) are in fact a fundamental class in the structure theorem of Geelen, Gerards and Whittle for the class of all $\mathbb{F}$-representable matroids. I think the term frame matroids is being floated around as a possible name (at least this is what Geelen called them during a talk this week). This sort of conflicts with some earlier naming conventions though. For more information, check out the Matroid Union Blog.


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