A new matroid lift construction and an application to group-labeled graphs

A well-known result of Brylawski constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. We generalize this result by showing how to construct a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of circuits of $M$. We conjecture that every lift of $M$ arises via this construction. We then apply this result to group-labeled graphs, generalizing a construction of Zaslavsky. Given a graph $G$ with edges labeled by a group, Zaslavsky's lift matroid $K$ is an elementary lift of the graphic matroid $M(G)$ that respects the group-labeling; specifically, the cycles of $G$ that are circuits of $K$ coincide with the cycles that are balanced with respect to the group-labeling. For $k \ge 2$, when does there exist a rank-$k$ lift of $M(G)$ that respects the group-labeling in this same sense? For abelian groups, we show that such a matroid exists if and only if the group is isomorphic to the additive group of a non-prime finite field.