Polymatroids are to finite groups as matroids are to finite fields

Abstract

Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} \Gamma_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $\Gamma_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$ a prime, and $b=p,$ then $P(\mathcal{H},b)$ is the usual matroid associated to any $\mathbb{Z}/p\mathbb{Z}$-matrix whose row space equals $\mathcal{H}.$ In general, there are many ways in which the relationship between $P(\mathcal{H},b)$ and $\mathcal{H}$ mirrors that of the relationship between a matroid and a subspace of a finite vector space. These include representability by excluded minors, the Crapo-Rota critical theorem, the existence of a concrete algebraic object representing the polymatroid dual of $P(\mathcal{H},b),$ analogs of Greene's theorem and the MacWilliams identities when $\mathcal{H}$ is a group code over a nonabelian group, and a connection to the combinatorial Laplacian of a quotient space determined by $\mathcal{G}$ and $\mathcal{H}.$ We use the group Crapo-Rota critical theorem to demonstrate an extension to hypergraphs of the classical duality between proper colorings and nowhere-zero flows on graphs.