Whitney numbers of partial dowling lattices

The Dowling lattice Q_n(G), G a finite group, generalizes the geometric lattice generated by all vectors, over a field, with at most two nonzero components. Abstractly, it is a fundamental object in the classification of finite matroids. Constructively, it is the frame matroid of a certain gain graph known as G K{_n}{^V}. Its Whitney numbers of the first kind enter into several important formulas. Ravagnani suggested and partially proved that these numbers of Q_n(G) and higher-weight generalizations are polynomial functions of |G|. We give a simple proof for Q_n(G) and its generalization to a wider class of gain graphs and biased graphs, and we determine the degrees and coefficients of the polynomials.

کلیدواژه ها:

Matroid ، Whitney number ، Dowling lattice ، group expansion gain graph