Dowling Geometry - Graphical Definitions

Graphical Definitions

A graphical definition was then given by Doubilet, Rota, and Stanley (1972). We give the slightly simpler (but essentially equivalent) graphical definition of Zaslavsky (1991), expressed in terms of gain graphs.

Take n vertices, and between each pair of vertices, v and w, take edges labelled by all possible elements of the group G. The edges are oriented, in that, if the label in the direction from v to w is the group element g, then the label of the same edge in the opposite direction, from w to v, is g−1. The label of an edge therefore depends on the direction of the edge; such labels are called gains. Also add to each vertex a loop whose gain is any value other than 1. (1 is the group identity element.) This gives a graph which is called GK_no (note the raised circle).

A cycle in the graph then has a gain. The cycle is a sequence of edges, e₁e₂···e_k. Suppose the gains of these edges, in a fixed direction around the cycle, are g₁, g₂, ..., g_k. Then the gain of the cycle is the product, g₁g₂···g_k. The value of this gain is not completely well defined, since it depends on the direction chosen for the cycle and on which is called the "first" edge of the cycle. What is independent of these choices is the answer to the following question: is the gain equal to 1 or not? If it equals 1 under one set of choices, then it is also equal to 1 under all sets of choices.

To define the Dowling geometry, we specify the circuits (minimal dependent sets). The circuits of the matroid are

• the cycles whose gain is 1,
• the pairs of cycles with both gains not equal to 1, and which intersect in a single vertex and nothing else, and
• the theta graphs in which none of the three cycles has gain equal to 1.

Thus, the Dowling geometry Q_n(G) is the frame matroid or (bias matroid) of the gain graph GK_no (the raised circle denotes the presence of loops). Other, equivalent definitions are described in the article on gain graphs.