On reading about matroids representable over partial fields, one learns about the 6th root of unity partial field, but other even-th root of unity partial fields seem to be absent from the standard repertoire of examples. Why is this? Is the matroid theory boring or not useful for those partial fields? I'd be particularly interested in knowing what is known about matroids over $\{1, -1, i, -i\}$ because it might be useful to me in another problem.


Thanks.



matroids representable over root of unity partial fields that are not the 6th root of unity partial field
Consider the partial field $P:=\{p\in \mathbb{C}: |p|=1\}$. It is a theorem of Whittle that if a matroid $M$ is representable over $P$, then $M$ is sixth-root of unity.


$P$ contains each $P\_k:=\{k^\text{th}\text{ roots}\}$, so all the $k$-root of unity matroids are also sixth-root-of-unity.


To represent $U\_{2,4}$ over $P\_k$ you need a $p\in P\_k$ so that $(1-p)\in P\_k$ - which happens only if $k$ is a multiple of $6$. Otherwise, the matroids over $P\_k$ are both binary and sixth-root, and hence regular. 


If you're looking for a really interesting class extending the sixth-root-of-unity matroids, I can recommend the 'quaternionic unimodular' matroids defined here: <https://arxiv.org/abs/1106.3088> 
They are pretty wild, but they should not be completely unmanageable: they do not contain $U\_{2,7}$ if I remember right.


It is a conjecture in that paper that quaternionic unimodular matroids have the half plane property.


