Whitney numbers of partial dowling lattices

Document Type : Research Paper

Author

Binghamton University, Binghamton, NY 13902-6000, U.S.A.

Abstract

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.

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References

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History

  • Receive Date: 13 September 2022
  • Revise Date: 22 May 2023
  • Accept Date: 28 May 2023
  • Published Online: 01 June 2024

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