Whitney Numbers of Partial Dowling Lattices

2022/09/05 by Thomas Zasĺavsky, Zaslavsky, Thomas
Mathematics · Computer Science · #Finite Group Theory Research #Advanced Combinatorial Mathematics #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2209.01775

Abstract

The Dowling lattice Q_n(\mathfrakG), \mathfrakG 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 \mathfrakG⋅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(\mathfrakG) and higher-weight generalizations are polynomial functions of |\mathfrakG|. We give a simple proof for Q_n(\mathfrakG) and its generalization to a wider class of gain graphs and biased graphs, and we determine the degrees and coefficients of the polynomials.

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