Geometric Lattices of Structured Partitions

*Research and preparation of this report were substantially assisted by National Science Foundation grants DMS-8407102 and DMS-8606102. For certain matroids on the edges of a graph, digraph, or bidirected graph with group gains (edge labels from a group), the ats form intriguing new geometric lattices whose Whitney numbers have geometrical signiicance. In this chapter, the rst of a series, we lay the theoretical foundation for the study of interesting special examples of graphical matroids and their lattices, including new general examples of geometric lattices based on groups and graphs, generalizing Dowling's lattices. (Indeed, fully half of our examples are join subsemilattices of Dowling lattices; the other half are a related but new type of geometric lattice.) We list some cryptomorphisms of the bias, lift, and complete lift matroids of a gain graph and use them to describe in detail the lattices of ats in terms of subgraphs and in terms of group-valued partitions. We also give methods for calculating chromatic invariants. From the examples in subsequent chapters we abstract the idea of a structured partition, concerning which we raise several questions for further research.