6 problems

Let $M$ be a matroid with circuit set $\mathcal{C}$, and suppose that an adjoint of $M$ exists. Let $\delta M^*$ denote the combinatorial derived matroid of the dual matroid. Dual-…

Fix a field $\mathbb{F}$ with algebraic closure $\overline{\mathbb{F}}$. Let $M$ be representable over $\mathbb{F}$, with $|E(M)|=n$ and $r(M)=k$, and let…

Let $M$ be a representable matroid with set of circuits $\mathcal{C}$. For matroids on a fixed ground set, use the weak order defined by $M_1\geq M_2$ when every dependent set of…

Let $M$ be a matroid with ground-set size $n$ and rank $k$, and let $M^*$ denote its dual. Assume that $M^*$ has an adjoint. Adjoint conjecture. Then $\delta M$ is isomorphic to an…

Let $M$ be a matroid with set of circuits $\mathcal{C}$. For $S\subseteq\mathcal{C}$, let $\operatorname{supp}(S)$ be the union of the circuits in $S$, and let $\mathcal{A}$ be the…

Let $R\in\mathbb{Z}^{k\times n}$ be sufficiently generic, so that $M(R)=U(k,n)$, and let $\delta_{OW}M(R)$ denote the Oxley–Wang derived matroid of the representation $R$. Generici…