Delta-matroids as subsystems of sequences of Higgs lifts

Delta-matroids generalize matroids. In a delta-matroid, the counterparts of bases, which are called feasible sets, can have different sizes, but they satisfy a similar exchange property in which symmetric differences replace set differences. One way to get a delta-matroid is to take a matroid L, a quotient Q of L, and all of the Higgs lifts of Q toward L; the union of the sets of bases of these Higgs lifts is the collection of feasible sets of a delta-matroid, which we call a full Higgs lift delta-matroid. We give an excluded-minor characterization of full Higgs lift delta-matroids within the class of all delta-matroids. We introduce a class of full Higgs lift delta-matroids that arise from lattice paths and that generalize lattice path matroids. It follows from results of Bouchet that all delta-matroids can be obtained from full Higgs lift delta-matroids by removing certain feasible sets; to address which feasible sets can be removed, we give an excluded-minor characterization of delta-matroids within the more general structure of set systems. This result in turn yields excluded-minor characterizations of a number of related classes of delta-matroids. (This is joint work with Carolyn Chun and Steve Noble.)