Determining all self-dual uniform matroids.

Question

I want to determine all self-dual uniform matroids; I know that the dual of a uniform matroid $U_{r,n}$ is $U_{n - r,n}$ by Example $2.1.4$ in James Oxley, second edition, "Matroid Theory". Which means that the maximal independent sets of the matroid become dependent sets of the dual matroid and the dependent sets ( I am not inclined to say the minimal dependent sets because I am not sure from this yet, if anyone clarify this point I would appreciate it) of the matroid become maximal independent sets of the dual. So, I think the set of all self-dual uniform matroids are the matroids that satisfy that $n - r = r$ or $n = 2r.$ Am I correct? How can I prove this?

Answer 1

You are correct that a a uniform matroid $U_{r,n}$ is self-dual if and only if $n=2r$ and the proof follows directly from Example 2.1.4 in Oxley, together with the facts that all uniform matroids on $n$ elements with rank $r$ are isomorphic, and that $U_{r,n}$ and $U_{r',n}$ are not isomorphic if $r \neq r'$.

However, you say "Which means that the maximal independent sets of the matroid become dependent sets of the dual matroid and the dependent sets of the matroid become maximal independent sets of the dual." This is not true. Note that since $U_{r,2r}$ is self dual its maximal independent sets are maximal independent sets in the dual as well. Given a matroid $M$ with ground set $X$, the dual matroid $M^*$ is the matroid with groundset $X$ such that a set $A \subseteq X$ is independent in $M^*$ if and only if $X-A$ contains a basis for $M$.